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'.2."@8 DC=40<5=B0;L=KE 2708<>459AB28O</a>&nbsp;|&nbsp; --> <a href="page09.html"><font size="-1">@8<5@K ?@>O2;5=8O BQ<=>9 M=5@388</a>&nbsp;|&nbsp; <a href="page08.html"><font size="-1">820O <5E0=8:0<font color="red" size="3"><sup>new</sup></font></a>&nbsp;|&nbsp; </td> <td>&nbsp;</td> </tr> </table> <h1 align="center" ><font color="#003366"><i>040G0 8>D0=B0 8 B5>@5<0 $5@<0</i></font></h1> </td> </tr> <tr ><font color="#003366"> <td>&nbsp;</td> <td td style="color: #000000" background="../mirgorodskiyal_image/skool.gif"> <h3 align="center">I.&nbsp;040G0 8>D0=B0</h3> <p> 1979 3>4C <=5 C40;>AL ?@8>1@5AB8 :=83C 2 @CAA:>< ?5@52>45 ""2>@FK <0B5<0B8:8" 8725AB=>3> 0<5@8:0=A:>3> 8AB>@8:0 <0B5<0B8:8 -@8:0 "5<?;0 5;;0, 8740==CN 2 !( 2 1937 3>4C&nbsp;<font size="2" color="#660000">(=83C 2 D>@<0B5 djvu <>6=> A:0G0BL&nbsp;<a href="http://math.ru/lib/author/?n=0&page=1&o_by=author&o=0">&nbsp;745AL&nbsp;</a>).</font>&nbsp;.  :=835 >48= 87 5Q >G5@:>2 ?>A2OIQ= L5@C $5@<0 :0: :>@>;N ;N18B5;59 <0B5<0B8:8. > CB25@645=8N -.". 5;;0, $5@<0 8<5; ?@82KG:C, G8B0O "@8D<5B8:C" 8>D0=B0 ;5:A0=4@89A:>3> 8 @07<KH;OO =04 5Q 7040G0<8, 70?8AK20BL =0 ?>;OE :=838 @57C;LB0BK A2>8E @07<KH;5=89 2 2845 :@0B:8E 70<5G0=89. 0 ?>;OE 2B>@>9 :=838 "@8D<5B8:8", :><<5=B8@CO 2>AL<CN 7040GC 8>D0=B0, $5@<0 =0?8A0;:</p> <p><i>"0>1>@>B, =52>7<>6=> @07;>68BL =8 :C1 =0 420 :C10, =8 18:204@0B =0 420 18:204@0B0, 8, 2>>1I5, =8:0:CN AB5?5=L, 1>;LHCN :204@0B0 =0 425 AB5?5=8 A B5< 65 ?>:070B5;5<. / >B:@K; MB><C ?>8AB8=5 GC45A=>5 4>:070B5;LAB2>, => MB8 ?>;O 4;O =53> A;8H:>< <0;K" /<i>-.".5;; «"2>@FK <0B5<0B8:8». .,1979, AB@.69</i>&nbsp;/.</i></p> <p> ><<5=B0@89 : 7040G5 8>D0=B0, :>340 >= 1K; >1=0@>4>20= ?>A;5 A<5@B8 $5@<0, ?>;CG8; =0720=85 <b>25;8:>9 B5>@5<K $5@<0</b>.  B0:>9 D>@<C;8@>2:5 <=5 4> MB>3> 2@5<5=8 >=0 =5 1K;0 8725AB=0.</p> <p> 07<KH;5=8O =04 ?@8<5G0=85< $5@<0 : 7040G5 8>D0=B0 8 @07<KH;5=8O =04 7040G59 8 5Q C@02=5=85< <5=O ?@825;8 : ?@54?>;>65=8N, GB> $5@<0 =5 =0?8A0; CA;>28O B5>@5<K 8 =0?8A0; ;8HL 5Q 70:;NG5=85, :>B>@>5 157 5Q CA;>28O 4>:070BL =52>7<>6=> 2 ?@8=F8?5. #A;>285 B5>@5<K >?@545;O5B :0: 5Q 70:;NG5=85, B0: 8 <5B>4 4>:070B5;LAB20. >-<>5<C, B5>@5<0 $5@<0 B0:>20: </p> <p><b><i>" A;8 C@02=5=85</i></b></p> <table border="0" width="100%"> <tr> <td>&nbsp;</td> <td align="center"><b><i>a <sup>2</sup> = x <sup>2</sup> + y <sup>2</sup></i></b></td> <td align="right">(1)</td> </tr> </table> <p><b><i>8<55B 15A:>=5G=>5 <=>65AB2> @5H5=89 =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5;, B>, =0>1>@>B, C@02=5=85</i></b></p> <table border="0" width="100%"> <tr> <td>&nbsp;</td> <td align="center"><b><i>x <sup>n</sup> + y <sup>n</sup> = z <sup>n</sup></i></b>, &nbsp;&nbsp;345 <b><i>n > 2</i></b></td> <td align="right">(2)</td> </tr> </table> <p><b><i>=5 8<55B @5H5=89 =0 <=>65AB25 F5;KE ?>;>68B5;L=KE G8A5;. "</i></b></p>. <p>>A;5 $5@<0 53> :><<5=B0@89 : 7040G5 8>D0=B0 =5 B>;L:> ?@8=O; D>@<C C@02=5=8O (2). >@074> 206=55 8 ACI5AB25==> B>, GB> >=> >:070;>AL ;8HQ==K< =5>1E>48<>9 A2O78 A C@02=5=85< (1) 8 ?@8=OBK< >4=>2@5<Q==> 70 CA;>285 8 70:;NG5=85 25;8:>9 B5>@5<K $5@<0. @C38E ?@8G8= <=>3>;5B=8E 157CA?5H=KE ?>8A:>2, O ?@54?>;>68;, GB> L5@ $5@<0 =5 =0?8A0; CA;>28O B5>@5<K, 0 =0?8A0; ;8HL 5Q 70:;NG5=85, :>B>@>5 157 CA;>28O B5>@5<K 4>:070BL =52>7<>6=>. =>9 ?@8G8=K <=>3>;5B=8E 157CA?5H=KE ?>8A:>2 4>:070B5;LAB20 C@02=5=8O (2) O =5 =0HQ;. :;NG5=85 C@02=5=8O (1) 2 25;8:CN B5>@5<C $5@<0 2 :0G5AB25 5Q CA;>28O =5 >B:@K20;> ?CBL : CA?5H=><C ?>8A:C 4>:070B5;LAB20 C@02=5=8O (2), 0 B@51>20;> 2 ?5@2CN >G5@54L 45B0;L=>3> 8AA;54>20=8O @5H5=89 C@02=5=8O (1) =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5;.</p> <p>AB>@8: <0B5<0B8:8 -.". 5;; ?@8=8<05B 2>AL<CN 7040GC 8>D0=B0 2 D>@<5 C@02=5=8O (1), 2 :>B>@>< :204@0BK 42CE ?5@5<5==KE <b><i>x <sup>2</sup></i></b> 8 <b><i>y <sup>2</sup></i></b> A;030NBAO 2 <b><i>a <sup>2</sup></i></b>, ?>A:>;L:C <K G8B05< A;520 =0?@02>. 0B5<0B8:8 4@52=8E =0@>4>2 G8B0;8 8 ?8A0;8 A?@020 =0;52> 8 ?@8 687=8 8>D0=B0 <>3;8 @0AA<0B@820BL 40==>5 C@02=5=85 A ?@>B82>?>;>6=>9 AB>@>=K:</p> <table border="0" width="100%"> <tr> <td>&nbsp;</td> <td align="center"><b><i>a <sup>2</sup> = x <sup>2</sup> + y <sup>2</sup> </i></b></td> <td align="right">(3)</td> </tr> </table> <p>N1>5 @5H5=85 C@02=5=8O (3) ?@54AB02;O5B A>1>9 <b><i>( a, x, y )</i></b> B@>9:0 ?8D03>@>2KE G8A5;. 0640O B@>9:0 ?8D03>@>2KE G8A5; ?@82O70=0 : A2>5<C >?@545;Q==><C =0BC@0;L=><C G8A;C, :>B>@>5 O2;O5BAO 8 5Q 8=48284C0;L=K< ?>@O4:>2K< =><5@><. 564C B@>9:>9 ?8D03>@>2KE G8A5; 8 5Q =><5@><, ?@8ACBAB2CNI8< 2 =0BC@0;L=>< @O4C G8A5;, ACI5AB2C5B 2708<=> >4=>7=0G=>5 A>>B25BAB285.</p> <p>"0:, 5A;8 40= 8=48284C0;L=K9 =><5@, B> ?> B@Q< D>@<C;0< <>6=> >?@545;8BL G8A;0 B@>9:8 <b><i>(a, x, y)</i></b>, 0 5A;8 40=0 B@>9:0 ?8D03>@>2KE G8A5;, B> <>6=> ?> B5< 65 D>@<C;0< >?@545;8BL 5Q 8=48284C0;L=K9 =><5@.</p> <p>5G5B=>5 =0BC@0;L=>5 G8A;> <b><i>N = 2k + 1</i></b> .  =5<C "?@82O70=0" B@>9:0 ?8D03>@>2KE G8A5;. '8A;0 B@>9:8 >?@545;ONBAO ?> D>@<C;0<: </p> <table border="0" width="100%"> <tr> <td width="30%" align="center"><font size="-1"><b><i>k = 1,&nbsp;&nbsp;2,&nbsp;&nbsp; 3,&nbsp;&nbsp;...</i></b></font></td> <td width="40%" align="center"><font size="-1"><b><i>N = 3,&nbsp;&nbsp;k = 1</i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i>(&nbsp;&nbsp;y = 3,&nbsp;&nbsp;x = 4,&nbsp;&nbsp;a = 5&nbsp;&nbsp;)</i></b></font></td> </tr> <tr> <td width="30%" align="center"><font size="-1"><b><i> &nbsp;y =&nbsp;&nbsp;N =&nbsp;&nbsp;2k&nbsp;&nbsp;+&nbsp;&nbsp;1&nbsp;</i></b></font></td> <td width="40%" align="center"><font size="-1"><b><i>N&nbsp;&nbsp;=5,&nbsp;&nbsp;k&nbsp;=&nbsp;2&nbsp;</i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i> (&nbsp;&nbsp;y&nbsp;&nbsp;=&nbsp;&nbsp;5,&nbsp;&nbsp;x&nbsp;&nbsp;=&nbsp;&nbsp;12,&nbsp;&nbsp;a&nbsp;&nbsp;=&nbsp;&nbsp;13&nbsp;&nbsp;)</i></b></font></td> </tr> <tr> <td width="30%" align="center"> <table border="0" height="100%"> <tr> <td align="center" height="49%"><font size="-1"><b><i>&nbsp;&nbsp;( 2k + 1 )<sup> 2 </sup>&nbsp;&nbsp;-&nbsp;&nbsp;1</i></b></font></td> </tr> <tr> <td align="center" height="1%"><img src="../mirgorodskiyal_image/line_black.gif" border="0" width="100" height="1"></td> </tr> <tr> <td align="center" height="50%"><b><i><font size="-1">2</font></i></b></td> </tr> </table></td> <td width="40%" align="center"><font size="-1"><b><i>N&nbsp;&nbsp;=&nbsp;&nbsp;7,&nbsp;&nbsp;k&nbsp;&nbsp;=&nbsp;&nbsp;3</i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i> (&nbsp;&nbsp;y&nbsp;&nbsp;=&nbsp;&nbsp;7,&nbsp;&nbsp;x&nbsp;&nbsp;=&nbsp;&nbsp;24,&nbsp;&nbsp;a&nbsp;&nbsp;=&nbsp;&nbsp;25&nbsp;&nbsp;)</i></b></font></td> </tr> <tr> <td width="30%" align="center"> <table border="0" height="100%"> <tr> <td align="center" height="49%"><font size="-1"><b><i>&nbsp;&nbsp;( 2k + 1 )<sup> 2 </sup>&nbsp;&nbsp;+&nbsp;&nbsp;1</i></b></font></td> </tr> <tr> <td align="center" height="1%"><img src="../mirgorodskiyal_image/line_black.gif" border="0" width="100" height="1"></td> </tr> <tr> <td align="center" height="50%"><b><i><font size="-1">2</font></i></b></td> </tr> </table></td> <td width="40%" align="center"><font size="-1"><b><i>N&nbsp;&nbsp;=&nbsp;&nbsp;9,&nbsp;&nbsp;k&nbsp;&nbsp;=&nbsp;&nbsp;4</i></b></font></td> <td width="30%" align="right"><font size="-1"><b><i>(&nbsp;&nbsp;y&nbsp;&nbsp;=&nbsp;&nbsp;9,&nbsp;&nbsp;x&nbsp;&nbsp;=&nbsp;&nbsp;40,&nbsp;&nbsp;a&nbsp;&nbsp;=&nbsp;&nbsp;41&nbsp;&nbsp;)</i></b></font></td> </tr> <tr> <td width="30%" align="center"><b><i>&nbsp;</i></b></td> <td width="40%" align="center"><font size="-1"><b><i>N&nbsp;&nbsp;=&nbsp;&nbsp;11,&nbsp;&nbsp;k&nbsp;&nbsp;=&nbsp;&nbsp;5</i></b></font></td> <td width="30%" align="right"><font size="-1"><b><i>(&nbsp;&nbsp;y&nbsp;&nbsp;=&nbsp;&nbsp;11,&nbsp;&nbsp;x&nbsp;&nbsp;=&nbsp;&nbsp;60,&nbsp;&nbsp;a&nbsp;&nbsp;=&nbsp;&nbsp;61&nbsp;&nbsp;)</i></b></font></td> </tr> <tr> <td>&nbsp;</td> <td>&nbsp;</td> <td align="center"><font size="-1"><b><i>&nbsp;&nbsp;8 &nbsp;&nbsp;B.&nbsp;&nbsp;4.</i></b></font></td> </tr> </table> <p>'QB=>5 G8A;> <b><i>N = 2k</i></b>.  =5<C "?@82O70=0" B@>9:0 ?8D03>@>2KE G8A5;. <b><i>( a, x, y )</i></b>, 5Q G8A;0 >?@545;ONBAO ?> D>@<C;0<: </p> <TABLE BORDER="0" WIDTH="100%"> <tr> <td width="30%" align="center"><font size="-1"><b><i>&nbsp;&nbsp;N &nbsp;&nbsp;=&nbsp;&nbsp;2 k, </i></b></font></td> <td width="40%" align="center"><font size="-1"><b><i>&nbsp;&nbsp;k&nbsp;&nbsp;=&nbsp;&nbsp;2,&nbsp;&nbsp;3,&nbsp;&nbsp;4,&nbsp;&nbsp;... </i></b></font></td> <td width="30%" align="center"><b><i>&nbsp;</i></b></td> </tr> <tr> <td width="30%" align="center"><font size="-1"><b><i>y = 4 k </i></b></font></td> <td width="40%" align="center"><font size="-1"><b><i>N = 4,&nbsp;&nbsp; k = 2</i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i> ( y = 8, &nbsp;&nbsp; x = 15,&nbsp;&nbsp; a = 17 ) </i></b></font></td> </tr> <tr> <td width="30%" align="center"><font size="-1"><b><i> x = 4 k <sup>2</sup> - 1</i></b></font></td> <td width="40%" align="center"><font size="-1"><b><i> N = 6,&nbsp;&nbsp; k = 3</i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i>( y = 12,&nbsp;&nbsp; x = 35, &nbsp;&nbsp; a = 37 )</i></b></font></td> </tr> <tr> <td width="30%" align="center"><b><i>&nbsp;&nbsp;</i></b></td> <td width="40%" align="center"><font size="-1"><b><i> N = 8,&nbsp;&nbsp; k = 4 </i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i>( y = 16,&nbsp;&nbsp; x = 63,&nbsp;&nbsp; a = 65 )</i></b></font></td> </tr> <tr> <td width="30%" align="center"><font size="-1"><b><i> a = 4 k <sup> 2 </sup> + 1 </i></b></font></td> <td width="40%" align="center"><font size="-1"><b><i> N = 10,&nbsp;&nbsp; k = 5 </i></b></font></td> <td width="30%" align="center"><font size="-1"><b><i> ( y = 99,&nbsp;&nbsp; x = 99, &nbsp;&nbsp;a = 101 ) </i></b></font></td> </tr> <tr> <td width="30%" align="center"><b><i>&nbsp;</i></b></td> <td width="40%" align="center"><b><i>&nbsp;</i></b></td> <td width="30%" align="center"><font size="-1"><b><i>&nbsp;&nbsp; 8 B. 4. &nbsp;&nbsp;</i></b></font></td> </tr> </table> <p>@><5 B@>5: ?8D03>@>2KE G8A5;, "?@82O70==KE" : A2>5<C 8=48284C0;L=><C =><5@C, ?> :>B>@><C >=8 >?@545;ONBAO C:070==K<8 D>@<C;0<8, ACI5AB2C5B 15A:>=5G=>5 <=>65AB2> B@>5: ?8D03>@>2KE G8A5;, "A2>1>4=KE" >B =><5@>2, "157=><5@=KE", :>B>@K5 =0E>4OBAO ?> 4@C38< D>@<C;0<. 0?@8<5@, B@>9:8 ?8D03>@>2KE G8A5; <b><i>( y = 20, x = 21, a = 29 ), ( y = 33, x = 56, a = 65 )</i></b> 8 4@C385 =5 8<5NB 8=48284C0;L=KE =><5@>2 8 4>?>;=ONB A>1>9 <=>65AB2> ?8D03>@>2KE G8A5;, 8<5NI8E 8=48284C0;L=K5 =><5@0.</p> <p>A;8 :064><C =0BC@0;L=><C G8A;C <b><i>N > 2</i></b> A>>B25BAB2C5B A2>O >?@545;5==0O B@>9:0 ?8D03>@>2KE G8A5;, 5A;8 8E AG5B=>5 <=>65AB2> >1J548=8BL A <=>65AB2>< "157=><5@=KE" ?8D03>@>2KE G8A5;, B> >1J548=Q==>5 <=>65AB2> ?8D03>@>2KE G8A5; 1C45B 70:;NG0BL 2 A515 =0BC@0;L=K9 @O4 G8A5; 2 :0G5AB25 ?>4<=>65AB20.</p> <p> C@02=5=88 (3) <b><i>a <sup>2</sup></i></b> ?>;CG05B A2>Q >?@545;5=85 8 2K@065=85 2 D>@<5 AC<<K ?5@5<5==KE <b><i>x <sup>2</sup></i></b> 8 <b><i>y <sup>2</sup></i></b> .</p> <p>@>872>;L=> 27OBK9 :204@0B G8A;0 <b><i>a <sup>2</sup></i></b> 2 ?@>F5AA5 @07;>65=8O =0 <b><i>x <sup>2</sup></i></b> 8 <b><i>y <sup>2</sup></i></b> <b><i>&nbsp;&nbsp;=5 87<5=O5B A2>53> G8A;>2>3> 7=0G5=8O</i></b>, 4>:07K20O A2>N =57028A8<>ABL >B G8A;>2KE 7=0G5=89 ?5@5<5==KE <b><i>x <sup>2</sup></i></b> 8 <b><i>y <sup>2</sup></i></b> .</p> <p>7<5=5=85 7=0G5=89 ?5@5<5==KE <b><i>x</i></b> 8 <b><i>y</i></b> =0E>48BAO 2 ?>4G8=Q==>< >B=>H5=88 : 7=0G5=8N ?>AB>O==>9 <b><i>a</i></b>. =8 ?@8=8<0NB 7=0G5=8O, :>B>@K5 <5=LH5 7=0G5=8O <b><i>a</i></b>, 8 B>;L:> B5 7=0G5=8O, ?@8 :>B>@KE AC<<0 8E :204@0B>2 @02=0 :204@0BC <b><i>a.</i></b> @C38<8 A;>20<8, >1;0ABL 4>?CAB8<KE 7=0G5=89 ?5@5<5==KE <b><i>x</i></b> 8 <b><i>y</i></b> >?@545;O5BAO 7=0G5=85< ?>AB>O==>9 <b><i>a :</i></b>&nbsp;&nbsp;&nbsp; <b><i>x < a</i></b>,&nbsp;&nbsp;&nbsp; <b><i>y < a</i></b>.</p> <p>0E>645=85 @5H5=89 C@02=5=8O (3), @02=>A8;L=>3> C@02=5=8N (1), A2>48BAO:</p> <font size="-1"><li>: =0E>645=8N 2A53> <=>65AB20 B@>5: $ +% '!,</li> <li>: =0E>645=8N B@>5: 25;8G8=  ?@O<>;8=59=KE >B@57:>2,</li> <li>: =0E>645=8N B@>5: 25;8G8= ) :204@0B>2,</li> <li>: =0E>645=8N B@>5: 25;8G8= * ?0@0;;5;5?8?54>2.</li></font> <p>> B5>@5<5 8D03>@0, ?;>I04L :204@0B0, ?>AB@>5==>3> =0 38?>B5=C75 <b><b>a</b></b> ?@O<>C3>;L=>3> B@5C3>;L=8:0, @02=0 AC<<5 ?;>I0459 :204@0B>2, ?>AB@>5==KE =0 53> :0B5B0E <b><i>x</i></b> 8 <b><i>y</i></b>. </p> <p><img src="../mirgorodskiyal_image/image077.gif" width="334" height="236" border="0" align="left" alt=" 8A. 1"> 564C G;5=0<8 C@02=5=8O (3) 8 ?;>I04O<8 :204@0B>2, ?>AB@>5==KE =0 AB>@>=0E ?@O<>C3>;L=>3> B@5C3>;L=8:0, ACI5AB2C5B 2708<=> >4=>7=0G=>5 A>>B25BAB285. >6=> A515 ?@54AB028BL, GB> B@>9:8 G8A5; C@02=5=8O (3) A1@>A8;8 A A51O D>@<C =0BC@0;L=KE G8A5; 8 ?@8=O;8 =0 A51O 35><5B@8G5A:CN D>@<C AB>@>= 8 ?;>I0459 :204@0B>2.</p> <p> 7 40==>3> CA;>28O A;54C5B 70:;NG5=85: <>6=> @07;>68BL ?;>I04L :204@0B0 =0 ?;>I048 42CE :204@0B>2.</p> <p>204@0BK <b><i>a <sup>2</sup></i></b> , <b><i>x <sup>2</sup></i></b> 8 <b><i>y <sup>2</sup></i></b> <>6=> @0AA<0B@820BL 2 B@QE<5@=>< ?@>AB@0=AB25 :0: :204@0B=K5 >A=>20=8O B@QE ?0@0;;5;5?8?54>2 >48=0:>2>9 ?@>872>;L=>9 2KA>BK <b><i>h</i></b>. </p> <p>#<=>68< >15 AB>@>=K C@02=5=8O (3) =0 F5;>5 G8A;> <b><i>h</i></b>.  @57C;LB0B5 ?>;CG05< C@02=5=85: </p> <table border="0" width="100%"> <tr> <td>&nbsp;</td> <td align="center"><b><i>a <sup>2</sup> h = x <sup>2</sup> h + y <sup>2</sup> h </i></b></td> <td align="right">(4)</td> </tr> </table> <p><img src="../mirgorodskiyal_image/image049.jpg" width="242" height="348" border="0" align="left" alt=" 8A. 2">#@02=5=85 (4) @02=>A8;L=> C@02=5=8N (3), B0: :0: 2A5 @5H5=8O C@02=5=8O (3) O2;ONBAO @5H5=8O<8 C@02=5=8O (4) 8 >1@0B=>. 0 ;52>9 AB>@>=5 C@02=5=8O (4), ?@>872545=85 <b><i>a<sup>2</sup>h</i></b> <>6=> @0AA<0B@820BL :0: >1JQ< ?0@0;;5;5?8?540 2KA>B>9 <b><i>h</i></b>, ?>AB@>5==>3> =0 :204@0B5 38?>B5=C7K <b><i>a</i></b>. KA>B0 <b><i>h</i></b> <>65B 8<5BL 7=0G5=85 ;N1>3> =0BC@0;L=>3> G8A;0, B0: GB> 25@E=55 >A=>20=85 ?0@0;;5;5?8?540 <>65B >:070BLAO 2KH5 >1;0:>2. ( 8A.2)</p> <p>0 ?@02>9 AB>@>=5 C@02=5=8O (4) ?@>872545=85 <b><i>x<sup>2</sup>h</i></b> <>6=> @0AA<0B@820BL :0: >1JQ< ?0@0;;5;5?8?540 2KA>B>9 <b><i>h</i></b>, ?>AB@>5==>3> =0 :204@0B5, ?@8=04;560I5< :0B5BC <b><i>x</i></b>. @>872545=85 <i><b>y<sup>2</sup>h</b></i> <>6=> @0AA<0B@820BL :0: >1JQ< ?0@0;;5;5?8?540 2KA>B>9 <b><i>h</i></b>, ?>AB@>5==>3> =0 :204@0B5, ?@8=04;560I5< :0B5BC <b><i>y</i></b>. ;>I04L >A=>20=8O ?0@0;;5;5?8?540 ;52>9 AB>@>=K C@02=5=8O @02=0 AC<<5 ?;>I0459 >A=>20=89 ?0@0;;5;5?8?54>2 ?@02>9 AB>@>=K C@02=5=8O.</p> <p>@8 @025=AB25 ?;>I048 <b><i>a<sup>2</sup></i></b> AC<<5 ?;>I0459 <b><i>x<sup>2</sup></i></b> 8 <b><i>y<sup>2</sup></i></b> 8 ?@8 @025=AB25 2KA>B <b><i>h</i></b> B@QE ?0@0;;5;5?8?54>2 >1JQ< ?0@0;;5;5?8?540, ?>AB@>5==>3> =0 :204@0B5, ?@8=04;560I5< 38?>B5=C75 <b><i>a</i></b>, @07;0305BAO =0 >1J5<K 42CE ?0@0;;5;5?8?54>2, ?>AB@>5==KE =0 :204@0B0E, ?@8=04;560I8E :0B5B0< <b><i>x</i></b> 8 <b><i>y</i></b>. !;54>20B5;L=>, C@02=5=85 (4) ?>:07K205B, GB> >1JQ< ?0@0;;5;5?8?540 <b><i>a<sup>2</sup>h</i></b> 53> ;52>9 AB>@>=K @07;0305BAO =0 >1JQ<K <b><i>x<sup>2</sup>h</i></b> 8 <b><i>y<sup>2</sup>h</i></b> 42CE ?0@0;;5;5?8?54>2 53> ?@02>9 AB>@>=K. KA>B0 <b><i>h</i></b> ?0@0;;5?8?54>2 <>65B 1KBL @02=0 ;N1><C F5;><C ?>;>68B5;L=><C G8A;C, 2:;NG0O: <b><i>h = a</i></b>, ?@8 :>B>@>< C@02=5=85 (4) >1@0I05BAO 2 @02=>A8;L=>5 C@02=5=85. </p> <img src="../mirgorodskiyal_image/image112.png" width="241" height="187" border="0" alt=" 8A.3" align="left"> <table border="0" width="50%"> <tr> <td>&nbsp;</td> <td><b><i>a <sup>3</sup> = x <sup>2</sup> a + x <sup>2</sup> a </i></b></td> <td align="right">(5)</td> </tr> </table> <p>7 C@02=5=8O (5) A;54C5B: :C1 <b><i>a</i></b> @07;0305BAO =0 420 ?0@0;;5;5?8?540 <b><i>x <sup>2</sup>a</i></b> 8 <b><i>y <sup>2</sup>a</i></b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p> <table border="0" align="center"> <tr> <td>&nbsp;</td> <td><img src="../mirgorodskiyal_image/pat_swirls.gif" width="560" height="14" border="0"></td> <td>&nbsp;</td> </tr> </table> <h3 align="center">II.&nbsp;>:070B5;LAB2> 25;8:>9 B5>@5<K $5@<0</h3> <p>!@02=8< :><<5=B0@89 $5@<0 : 7040G5 8>D0=B0 A A>2@5<5==>9 D>@<C;8@>2:>9 25;8:>9 B5>@5<K $5@<0, 8<5NI59 284 C@02=5=8O. <br>«<em><b>#@02=5=85</em> <div align="center"><em>x<sup>n </sup>+ y<sup>n </sup>= z<sup>n</sup></em> (345 n  F5;>5 G8A;> 1>;LH55 42CE)</div><em>=5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E</b></em>»</p> <p>><<5=B0@89 =0E>48BAO A 7040G59 2 ;>38G5A:>9 A2O78, 0=0;>38G=>9 ;>38G5A:>9 A2O78 A:07C5<>3> A ?>4;560I8<. ">, GB> CB25@6405BAO 7040G59 8>D0=B0, =0>1>@>B CB25@6405BAO :><<5=B0@85< $5@<0.</p> <p>><<5=B0@89 $5@<0 <>6=> B0: B@0:B>20BL: 5A;8 :204@0B=>5 C@02=5=85 A B@5<O =58725AB=K<8 8<55B 15A:>=5G=>5 <=>65AB2> @5H5=89 =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5;, B>, =0>1>@>B, C@02=5=85 A B@5<O =58725AB=K<8 2 AB5?5=8, 1>;LH59 :204@0B0, =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E.</p> <p> C@02=5=88 =5B 4065 =0<5:0 =0 53> A2O7L A 7040G59 8>D0=B0. 3> CB25@645=85 B@51C5B 4>:070B5;LAB20, => ?@8 =Q< =5B CA;>28O, 87 :>B>@>3> A;54C5B, GB> >=> =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E.</p> <p>725AB=K5 <=5 20@80=BK 4>:070B5;LAB20 C@02=5=8O A2>4OBAO : A;54CNI5<C 0;3>@8B<C. <ol> <li>#@02=5=85 B5>@5<K $5@<0 ?@8=8<05BAO 70 5Q 70:;NG5=85, 2 A?@0254;82>AB8 :>B>@>3> C15640NBAO ?@8 ?><>I8 4>:070B5;LAB20.</li> <li>-B> 65 C@02=5=85 =07K20NB <strong>8AE>4=K<</strong> C@02=5=85<, 87 :>B>@>3> 4>;6=> 8AE>48BL 53> 4>:070B5;LAB2>.</li></p> <p> @57C;LB0B5 >1@07>20;0AL B02B>;>38O: «<em>A;8 C@02=5=85 =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E, B> >=> =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E</em>».>:070B5;LAB2> B02B>;>388 70254><> O2;O5BAO =5?@028;L=K< 8 ;8H5==K< 2AO:>3> A<KA;0. > 5Q 4>:07K20NB <5B>4>< >B ?@>B82=>3>.</p> <li>@8=8<05BAO ?@54?>;>65=85, ?@>B82>?>;>6=>5 B><C, GB> CB25@6405BAO C@02=5=85<, :>B>@>5 B@51C5BAO 4>:070BL. => =5 4>;6=> ?@>B82>@5G8BL 8AE>4=><C C@02=5=8N, 0 >=> 5<C ?@>B82>@5G8B. >:07K20BL B>, GB> ?@8=OB> 157 4>:070B5;LAB20, 8 ?@8=8<0BL 157 4>:070B5;LAB20 B>, GB> B@51C5BAO 4>:070BL, =5 8<55B A<KA;0.</li> <li>0 >A=>20=88 ?@8=OB>3> ?@54?>;>65=8O 2K?>;=ONBAO 01A>;NB=> ?@028;L=K5 <0B5<0B8G5A:85 >?5@0F88 8 459AB28O, GB>1K 4>:070BL, GB> >=> ?@>B82>@5G8B 8AE>4=><C C@02=5=8N 8 O2;O5BAO ;>6=K<.</li></ol> <p>>MB><C 2>B C65 370 ;5B 4>:070B5;LAB2> C@02=5=8O 25;8:>9 B5>@5<K $5@<0 >AB0QBAO =5>ACI5AB28<>9 <5GB>9 A?5F80;8AB>2 8 ;N18B5;59 <0B5<0B8:8.</p> <p><font size="3" face="Arial Narrow" color="#003366" weight="800"><u><i>/ ?@8=O; C@02=5=85 70 70:;NG5=85 B5>@5<K, 0 2>AL<CN 7040GC 8>D0=B0 8 5Q C@02=5=85 - 70 CA;>285 B5>@5<K.</i></u></font></p> <strong><div align="left">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;"5>@5<0.</div> <br>&quot;A;8 C@02=5=85 <b><i>x <sup> 2 </sup> + y <sup>2 </sup>= z <sup>2</sup></i></b>&nbsp;&nbsp;(1)&nbsp; 8<55B 15A:>=5G=>5 <=>65AB2> @5H5=89 =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5;, B>, =0>1>@>B, C@02=5=85 <b><i>x <sup>n </sup> + y <sup>n </sup> = z <sup>n</sup></i></b>, 345 <b><i>n &gt; 2</i></b>&nbsp;&nbsp;(2)&nbsp;=5 8<55B @5H5=89 =0 <=>65AB25 F5;KE ?>;>68B5;L=KE G8A5;.&quot;</strong> <p><strong><div align="left">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;>:070B5;LAB2>.</div></strong></p> <p><strong>)</strong> A5< 8725AB=>, GB> C@02=5=85 (1) 8<55B 15A:>=5G=>5 <=>65AB2> @5H5=89 =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5;. >:065<, GB> =8 >4=0 B@>9:0 ?8D03>@>2KE G8A5;, O2;ONI0OAO @5H5=85< C@02=5=8O (1), =5 O2;O5BAO @5H5=85< C@02=5=8O (2).</p> <p>0 >A=>20=88 70:>=0 >1@0B8<>AB8 @025=AB20, AB>@>=K C@02=5=8O (1) ?><5=O5< <5AB0<8. 8D03>@>2K G8A;0 <b><i>(<em>z, E, C</em>)</i></b> <>3CB 1KBL 8AB>;:>20=K :0: 4;8=K AB>@>= ?@O<>C3>;L=>3> B@5C3>;L=8:0, 0 :204@0BK <b><i>(<em> x <sup>2</sup>, y <sup>2</sup>, z <sup>2</sup></em>)</i></b> <>3CB 1KBL 8AB>;:>20=K :0: ?;>I048 :204@0B>2, ?>AB@>5==KE =0 53> 38?>B5=C75 8 :0B5B0E.</p> <img src="../mirgorodskiyal_image/ris1.jpg" width="285" height="221" align="left"> <p>;>I048 :204@0B>2 C@02=5=8O (1) C<=>68< =0 ?@>872>;L=CN 2KA>BC <b><i>h</i></b>:</p><br> <div align="center"><b><i><em>z <sup>2</sup>h = x <sup>2</sup>h + y <sup>2</sup> h</em></i></b>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3)</div> <p>#@02=5=85 (3) <>6=> B@0:B>20BL :0: @025=AB2> >1J5<0 ?0@0;;5;5?8?540 AC<<5 >1JQ<>2 42CE ?0@0;;5;5?8?54>2.</p> <p>CABL 2KA>B0 B@5E ?0@0;;5;5?8?54>2 <b><i>h = z</i></b>: <div align="center"><b><i><em>z <sup>3</sup> = x <sup>2</sup>z + y <sup>2</sup>z&nbsp;</em></i></b>&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(4)</div></p> <p>1J5< :C10 @07;>68;AO =0 420 >1J5<0 42CE ?0@0;;5;5?8?54>2. 1JQ< :C10 >AB028< 157 87<5=5=89, 0 2KA>BC ?5@2>3> ?0@0;;5;5?8?54 C<5=LH8< 4> <b><i>x</i></b> 8 2KA>BC 2B>@>3> ?0@0;;5;5?8?540 C<5=LH8< 4> <b><i>y</i></b>. 1JQ< :C10 1>;LH5 AC<<K >1JQ<>2 42CE :C1>2:</p> <div align="center"><b><i><em>z <sup>3</sup> &gt; x <sup>3</sup> + y <sup>3</sup> </em></i></b>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (5)</div> <p>0 <=>65AB25 B@>5: ?8D03>@>2KE G8A5; (<em><b>E, C, z</b></em>) ?@8 <b><i>n = 3</i></b> =5 <>65B 1KBL =8 >4=>3> @5H5=8O C@02=5=8O (2). !;54>20B5;L=>, =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5; =52>7<>6=> :C1 @07;>68BL =0 420 :C10.</p> <p>CABL 2 C@02=5=88 (3) 2KA>B0 B@QE ?0@0;;5;5?8?54>2 <b><em>h = z <sup>2</sup></em></b>:</p> <div align="center"><b><em>z <sup>2</sup>z <sup>2</sup> = x <sup>2</sup>z <sup>2</sup> + y <sup>2</sup>z <sup>2</sup></em></b>&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;(6)</div> <p>1J5< ?0@0;;5;5?8?540 @07;>68;AO =0 AC<<C >1JQ<>2 42CE ?0@0;;5;5?8?54>2. <br>52CN AB>@>=C C@02=5=8O (7) >AB028< 157 87<5=5=8O. 0 ?@02>9 53> AB>@>=5 2KA>BC <b><i><em>z <sup>2</sup></em></i></b> C<5=LH8< 4> <b><i>E</i></b> 2 ?5@2>< A;0305<>< 8 4> <b><i><em>C <sup>2</sup></em></i></b> 2> 2B>@>< A;0305<><.</p> <p>#@02=5=85 (7) >1@0B8;>AL 2 =5@025=AB2>:</p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%"><b><i><em>z <sup>4</sup> &gt; x <sup>4</sup> + y <sup>4</sup></em></i></b></td> <td width="10%">(7)</td> </tr> </table> <p>0 <=>65AB25 B@>5: ?8D03>@>2KE G8A5; ?@8 n=4 =5 <>65B 1KBL =8 >4=>3> @5H5=8O C@02=5=8O (2). !;54>20B5;L=>, =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5; =52>7<>6=> 18:204@0B @07;>68BL =0 420 18:204@0B0.</p> <p>CABL 2 C@02=5=88 (3) 2KA>BK ?0@0;;5;5?8?54>2 <b><i><font face="cursive" size="2">h = z <sup>n-2</sup></font></i></b></p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%"><b><i><em>z <sup>n</sup> = x <sup>2</sup>z <sup>n-2</sup> + y <sup>2</sup>z <sup>n-2</sup></em></i></b></td> <td width="10%" align="right">(8)</td> </tr> </table> <p>1J5< ?0@0;;5;5?8?540 @07;>68;AO =0 420 >1J5<0 42CE ?0@0;;5;5?8?54>2.</p> <br>52CN AB>@>=C C@02=5=8O (8) >AB028< 157 87<5=5=8O. <br>0 ?@02>9 AB>@>=5 2KA>BC <b><i>z <sup>n-2</sup></i></b> C<5=LH8< 4> <b><i>x <sup>n-2</sup></i></b> 2 ?5@2>< A;0305<>< 8 C<5=LH8< 4> <b><i>y <sup>n-2</sup></i></b> 2> 2B>@>< A;0305<><. #@02=5=85 (8) >1@0I05BAO 2 =5@025=AB2>: <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%"><b><i>z <sup>n</sup> &gt; x <sup>n</sup> + y <sup>n </sup></i></b></td> <td width="10%">&nbsp;(9)</td> </tr> </table> <p>0 <=>65AB25 B@>5: ?8D03>@>2KE G8A5; =5 <>65B 1KBL =8 >4=>3> @5H5=8O C@02=5=8O (2).</p> <p>!;54>20B5;L=>, =0 <=>65AB25 2A5E B@>5: ?8D03>@>2KE G8A5; ?@8 2A5E <b><i>n &gt; 2</i></b> C@02=5=85 (2) =5 8<55B @5H5=89.</p> <p>>;CG5=> "?>AB8=5 GC45A=>5 4>:070B5;LAB2>", => B>;L:> 4;O B@>5: <b>?8D03>@>2KE G8A5;</b>.  MB>< 70:;NG05BAO <b>=54>AB0B>: 4>:070B5;LAB20</b> 8 ?@8G8=0 >B:070 . $5@<0 >B =53>.</p> <p><b>B)</b>&nbsp;>:065<, GB> C@02=5=85 (2) =5 8<55B @5H5=89 =0 <=>65AB25 B@>5: =5?8D03>@>2KE G8A5;, ?@54AB02;ONI5< A1>9 A5<59AB2> ?@>872>;L=> 27OB>9 B@>9:8 ?8D03>@>2KE G8A5; <b><i>z = 13, x = 12, y = 5</i></b> 8 A5<59AB2> ?@>872>;L=> 27OB>9 B@>9:8 F5;KE ?>;>68B5;L=KE G8A5; <b><i>z = 21, x = 19, y = 16</i></b></p> <p>15 B@>9:8 G8A5; O2;ONBAO G;5=0<8 A2>8E A5<59AB2:</p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">(13, 12, 12); (13, 12,11);...; <b>(13, 12, 5)</b>;...; (13,7, 1);...; (13,1, 1)</td> <td width="10%" align="right">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(10)</td> </tr> <tr> <td width="10%"></td> <td width="80%">(21, 20, 20); (21, 20, 19);...;(21, 19, 16);...;(21, 1, 1)</td> <td width="10%" align="right">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(11)</td> </tr> </table> <p>'8A;> G;5=>2 A5<59AB20 (10) 8 (11) @02=> ?>;>28=5 ?@>872545=8O 13 =0 12 8 21 =0 20, B. 5. 78 8 210.</p> <p> :064>< G;5=5 A5<59AB20 (10) ?@8ACBAB2C5B <b><i>z = 13</i></b> 8 ?5@5<5==K5 <b><i>E</i></b> 8 <b><i>C</i></b>, :>B>@K5 ?@8=8<0NB 7=0G5=8O F5;KE G8A5; <b><i>13 &gt; x &gt; 0</i></b> , <b><i>13 &gt; y &gt; 0</i></b>. 5@5<5==K5 ?>A;54>20B5;L=> C1K20NB =0 <b><i>1</i></b></p>. <p> :064>< G;5=5 A5<59AB20 (11) ?@8ACBAB2C5B <b><i>z = 21</i></b> 8 ?5@5<5==K5 <b><i>E</i></b> 8 <b><i>C</i></b>, :>B>@K5 ?@8=8<0NB 7=0G5=8O F5;KE G8A5; <b><i>21 &gt; x &gt;0</i></b>, <b><i>21 &gt; y &gt; 0</i></b>. 5@5<5==K5 ?>A;54>20B5;L=> C1K20NB =0 <b><i>1</i></b>.</p> <p>"@>9:8 G8A5; ?>A;54>20B5;L=>AB8 (10) 8 (11) <>6=> ?@54AB028BL 2 2845 ?>A;54>20B5;L=>AB8 =5@025=AB2 B@5BL59 AB5?5=8: </p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">13<sup>3</sup> &lt; 12 <sup>3</sup> + 12 <sup>3</sup> ;13 <sup>3</sup> &lt; 12 <sup>3</sup> + 11 <sup>3</sup> ;...; 13 <sup>3</sup> &lt; 12 <sup>3</sup> + 8 <sup>3</sup>; 13 <sup>3</sup> &gt; 12 <sup>3</sup> + 7 <sup>3</sup> ;...; 13 <sup>3</sup> &gt; 1 <sup>3</sup> + 1 <sup>3</sup></td> <td width="10%">&nbsp;</td> </tr> <tr> <td width="10%">&nbsp;</td> <td width="80%">21 <sup>3</sup> &lt; 20 <sup>3</sup> + 20 <sup>3</sup> ; 21 <sup>3</sup> &lt; 20 <sup>3</sup> + 19 <sup>3</sup> ; ...; 21 <sup>3</sup> &lt; 19 <sup>3</sup> + 14 <sup>3</sup> ; 21 <sup>3</sup> &gt; 19 <sup>3</sup> + 13 <sup>3</sup> ;...; 21 <sup>3</sup> &gt; 1 <sup>3</sup> + 1<sup>3</sup></td> <td width="10%">&nbsp;</td> </tr> </table> <br>8 2 2845 =5@025=AB2 G5B25@B>9 AB5?5=8: <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">13 <sup>4</sup> &lt; 12 <sup>4</sup> + 12 <sup>4</sup> ;...; 13 <sup>4</sup> &lt; 12 <sup>4</sup> + 10 <sup>4</sup> ; 13 <sup>4</sup> &gt; 12 <sup>4</sup> + 9 <sup>4</sup> ;...; 13 <sup>4</sup> &gt; 1 <sup>4</sup> + 1 <sup>4</sup></td> <td width="10%">&nbsp;</td> </tr> <tr> <td width="10%">&nbsp;</td> <td width="80%">21 <sup>4</sup> &lt; 20 <sup>4</sup> + 20 <sup>4</sup> ; 21 <sup>4</sup> &lt; 20 <sup>4</sup> + 19 <sup>4</sup> ; ...; 21 <sup>4</sup> &lt; 19 <sup>4</sup> + 16 <sup>4</sup> ;& ; 21 <sup>4</sup> &gt; 1 <sup>4</sup> + 1 <sup>4</sup></td> <td width="10%">&nbsp;</td> </tr> </table> <p>@028;L=>ABL :064>3> =5@025=AB20 C4>AB>25@O5BAO 2>72KH5=85< G8A5; 2 B@5BLN 8 2 G5B25@BCN AB5?5=L.</p> <p>C1 1>;LH53> G8A;0 =52>7<>6=> @07;>68BL =0 420 :C10 <5=LH8E G8A5;. = 8;8 <5=LH5, 8;8 1>;LH5, AC<<K :C1>2 42CE <5=LH8E G8A5;.</p> <p>8:204@0B 1>;LH53> G8A;0 =52>7<>6=> @07;>68BL =0 420 18:204@0B0 <5=LH8E G8A5;. = 8;8 <5=LH5, 8;8 1>;LH5, AC<<K 18:204@0B>2 <5=LH8E G8A5;.</p> <p>! 2>7@0AB0=85< ?>:070B5;O AB5?5=8 2A5 =5@025=AB20, :@><5 ;52>3> :@09=53> =5@025=AB20, 8<5NB >48=0:>2K9 A<KA;:</p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">13 <sup>8</sup> &lt; 12 <sup>8</sup> + 12 <sup>8</sup> ; 13 <sup>8</sup> &gt; 12 <sup>8</sup> + 11 <sup>8</sup> ;...; 13 <sup>8</sup> &gt; 12 <sup>8</sup> + 5 <sup>8</sup> ;...; 13 <sup>8</sup> &gt; 1 <sup>8</sup> + 1 <sup>8</sup></td> <td width="10%">&nbsp;</td> </tr> <tr> <td width="10%">&nbsp;</td> <td width="80%">21 <sup>14</sup> &lt; 20 <sup>14</sup> + 20 <sup>14</sup> ; 21 <sup>14</sup> &gt; 20 <sup>14</sup> + 19 <sup>14</sup> ; ...; 21 <sup>14</sup> &gt; 19 <sup>14</sup> + 16 <sup>14</sup> ; ...; 21 <sup>14</sup> &gt; 1 <sup>14</sup> + 1 <sup>14</sup></td> <td width="10%">&nbsp;</td> </tr> </table> <p>@8 2>7@0AB0=88 ?>:070B5;O AB5?5=8 5I5 =0 1 2A5 =5@025=AB20, 2:;NG0O 8 ;52K5 :@09=85 =5@025=AB20, 8<5NB >48=0:>2K9 A<KA;:</p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">13 <sup>9</sup> &gt; 12 <sup>9</sup> + 12 <sup>9</sup> ; &gt; 12 <sup>9</sup> + 11 <sup>9</sup> ;& ; &gt; 12 <sup>9</sup> + 5 <sup>9</sup> ;& ; &gt; 1 <sup>9</sup> + 1 <sup>9</sup></td> <td width="10%">&nbsp;</td> </tr> <tr> <td width="10%">&nbsp;</td> <td width="80%">21 <sup>15</sup> &gt; 20 <sup>15</sup> + 20 <sup>15</sup> ; 21 <sup>15</sup> &gt; 20 <sup>15</sup> + 19 <sup>15</sup> ; ...; 21 <sup>15</sup> &gt; 19 <sup>15</sup> + 16 <sup>15</sup> ; ...; 21 <sup>15</sup> &gt; 1 <sup>15</sup> + 1 <sup>15</sup></td> <td width="10%">&nbsp;</td> </tr> </table> <p>@8 40;L=59H5< 2>7@0AB0=88 ?>:070B5;O AB5?5=8 =5@025=AB2 >=8 2A5 8<5NB >48=0:>2K9 A<KA;: AB5?5=L 1>;LH53> G8A;0 1>;LH5 AC<<K AB5?5=59 <5=LH8E 42CE G8A5; A B5< 65 ?>:070B5;5<:</p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">13 <sup>n</sup> &gt; 12 <sup>n</sup> + 12 <sup>n</sup> ; 13 <sup>n</sup> &gt; 12 <sup>n</sup> + 11 <sup>n</sup> ;...; 13 <sup>n</sup> &gt; 7 <sup>n</sup> + 4 <sup>n</sup> ;...; 13 <sup>n</sup> &gt; 1 <sup>n</sup> + 1 <sup>n</sup></td> <td width="10%">&nbsp;&nbsp;&nbsp;(12)</td> </tr> <tr> <td width="10%">&nbsp;</td> <td width="80%">21 <sup>n</sup> &gt; 20 <sup>n</sup> + 20 <sup>n</sup> ; 21 <sup>n</sup> &gt; 20 <sup>n</sup> + 19 <sup>n</sup> ;...; ;...; 21 <sup>n</sup> &gt; 1 <sup>n</sup> + 1 <sup>n</sup></td> <td width="10%">&nbsp;&nbsp;&nbsp;(13)</td> </tr> </table> <p>52K9 :@09=89 G;5= ?>A;54>20B5;L=>AB59 (12) (13) ?@54AB02;O5B A>1>9 =081>;55 A;01>5 =5@025=AB2>. 3> ?@028;L=>ABL >?@545;O5B ?@028;L=>ABL 2A5E ?>A;54CNI8E =5@025=AB2 ?>A;54>20B5;L=>AB8 (12) ?@8 <b><i>n &gt; 8</i></b> 8 ?>A;54>20B5;L=>AB8 (13) ?@8 <b><i>n &gt; 14</i></b>.</p> <p>!@548 =8E =5 <>65B 1KB =8 >4=>3> @025=AB20. @>872>;L=> 27OB0O B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; (21,19,16) =5 O2;O5BAO @5H5=85< C@02=5=8O (2) 25;8:>9 B5>@5<K $5@<0. A;8 ?@>872>;L=> 27OB0O B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; =5 O2;O5BAO @5H5=85< C@02=5=8O, B> C@02=5=85 =5 8<55B @5H5=89 =0 <=>65AB25 F5;KE ?>;>68B5;L=KE G8A5;, GB> 8 B@51>20;>AL 4>:070BL.</p> <p><b>!)</b>&nbsp;&nbsp; :><<5=B0@88 $5@<0 : 7040G5 8>D0=B0 CB25@6405BAO, GB> =52>7<>6=> @07;>68BL «<em>2>>1I5, =8:0:CN AB5?5=L, 1>;LHCN :204@0B0, =0 425 AB5?5=8 A B5< 65 ?>:070B5;5<</em>».</p> <p><strong>&5;CN</strong> AB5?5=L, 1>;LHCN :204@0B0, 459AB28B5;L=> =52>7<>6=> @07;>68BL =0 425 AB5?5=8 A B5< 65 ?>:070B5;5<. <strong>5F5;CN</strong> AB5?5=L, 1>;LHCN :204@0B0 <>6=> @07;>68BL =0 425 AB5?5=8 A B5< 65 ?>:070B5;5<.</p> <p>N10O ?@>872>;L=> 27OB0O B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; <b><i>(z, x, y)</i></b> <>65B ?@8=04;560BL A5<59AB2C, :064K9 G;5= :>B>@>3> A>AB>8B 87 ?>AB>O==>3> G8A;0 <b><i>z</i></b> 8 42CE G8A5;, <5=LH8E <b><i>z</i></b>. 064K9 G;5= A5<59AB20 <>65B 1KBL ?@54AB02;5= 2 D>@<5 =5@025=AB20, 0 2A5 ?>;CG5==K5 =5@025=AB20 - 2 2845 ?>A;54>20B5;L=>AB8 =5@025=AB2:</p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%" align="center"><b><i>  z<sup>n</sup> < (z - 1)<sup>n</sup> + (z - 1)<sup>n</sup>; &nbsp;z<sup>n</sup> < (z - 1)<sup>n</sup> + (z - 2)<sup>n</sup>;&nbsp; & ; &nbsp; z<sup>n</sup> > 1<sup>n</sup> + 1<sup>n</sup> </i></b></td> <td width="10%">&nbsp;&nbsp;&nbsp;&nbsp;(14)</td> </tr> </table> <p>>A;54>20B5;L=>ABL =5@025=AB2 (14) =0G8=05BAO =5@025=AB20<8, C :>B>@KE ;520O AB>@>=0 <5=LH5 ?@02>9 AB>@>=K, 0 >:0=G8205BAO =5@025=AB20<8, C :>B>@KE ?@020O AB>@>=0 <5=LH5 ;52>9 AB>@>=K. ! 2>7@0AB0=85< ?>:070B5;O AB5?5=8 <b><i>n > 2</i></b> G8A;> =5@025=AB2 ?@02>9 AB>@>=K ?>A;54>20B5;L=>AB8 (14) C25;8G8205BAO. @8 ?>:070B5;5 AB5?5=8 <b><i>n = k</i></b> 2A5 =5@025=AB20 ;52>9 AB>@>=K ?>A;54>20B5;L=>AB8 87<5=ONB A2>9 A<KA; 8 ?@8=8<0NB A<KA; =5@025=AB2 ?@02>9 AB>@>=K =5@025=AB2 ?>A;54>20B5;L=>AB8 (14).  @57C;LB0B5 2>7@0AB0=8O ?>:070B5;O AB5?5=8 C 2A5E =5@025=AB2 ;520O AB>@>=0 >:07K205BAO 1>;LH5 ?@02>9 AB>@>=K: </p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%">z <sup>k </sup>&gt; (z-1) <sup>k</sup> + (z-1) <sup>k</sup> ; z <sup>k</sup> &gt; (z-1) <sup>k</sup> + (z-2) <sup>k</sup> ;...; z <sup>k</sup> &gt; 2 <sup>k</sup> + 1 <sup>k</sup>; z <sup>k</sup> &gt; 1 <sup>k</sup> + 1 <sup>k</sup></td> <td width="10%">&nbsp;&nbsp;&nbsp;&nbsp;(15)</td> </tr> </table> <p>@8 40;L=59H5< 2>7@0AB0=88 ?>:070B5;O AB5?5=8 <b><i>n > k</i></b> =8 >4=> 87 =5@025=AB2 =5 87<5=O5B A2>53> A<KA;0 8 =5 >1@0I05BAO 2 @025=AB2>. 0 MB>< >A=>20=88 <>6=> CB25@640BL, GB> ;N10O ?@>872>;L=> 27OB0O B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; <b><i>(z, x, y)</i></b> ?@8 <b><i>n > 2</i></b> , <b><i>z > x</i></b>, <b><i>z > y</i></b> =5 O2;O5BAO @5H5=85< C@02=5=8O 1>;LH>9 B5>@5<K $5@<0. </p> <p> ?@>872>;L=> 27OB>9 B@>9:5 F5;KE ?>;>68B5;L=KE G8A5; <b><i>z</i></b> <>65B 1KBL A:>;L C3>4=> 1>;LH8< =0BC@0;L=K< G8A;><. ;O 2A5E =0BC@0;L=KE G8A5;, :>B>@K5 =5 1>;LH5 <b><i>z</i></b>, 1>;LH0O B5>@5<0 $5@<0 4>:070=0. </p> <p><b>D)</b> 0:8< 1K =8 1K;> 1>;LH8< G8A;> <b><i>z</i></b>, 2 =0BC@0;L=>< @O4C G8A5; 4> =53> 8<55BAO 1>;LH>5, => :>=5G=>5 <=>65AB2> F5;KE G8A5;, 0 ?>A;5 =53>  15A:>=5G=>5 <=>65AB2> F5;KE G8A5;. </p> <p>>:065<, GB> 2A5 15A:>=5G=>5 <=>65AB2> =0BC@0;L=KE G8A5;, 1>;LH8E <b><i>z</i></b>, >1@07CNB B@>9:8 G8A5;, :>B>@K5 =5 O2;ONBAO @5H5=8O<8 C@02=5=8O 1>;LH>9 B5>@5<K $5@<0, =0?@8<5@, ?@>872>;L=> 27OB0O B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; <b><i>(z + 1, x ,y)</i></b>, 2 :>B>@>9 <b><i>z + 1 > x</i></b> 8 <b><i>z + 1 > y</i></b> ?@8 2A5E 7=0G5=8OE ?>:070B5;O AB5?5=8 <b><i>n > 2</i></b> =5 O2;O5BAO @5H5=85< C@02=5=8O 1>;LH>9 B5>@5<K $5@<0. </p> <p>@>872>;L=> 27OB0O B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; <b><i>(z + 1, x, y)</i></b> <>65B ?@8=04;560BL A5<59AB2C B@>5: G8A5;, :064K9 G;5= :>B>@>3> A>AB>OB 87 ?>AB>O==>3> G8A;0 <b><i>z + 1</i></b> 8 42CE G8A5; <b><i>E</i></b> 8 <b><i>C</i></b>, ?@8=8<0NI8E @07;8G=K5 7=0G5=8O, <5=LH85 <b><i>z + 1</i></b>. ';5=K A5<59AB20 <>3CB 1KBL ?@54AB02;5=K 2 D>@<5 =5@025=AB2, C :>B>@KE ?>AB>O==0O ;520O AB>@>=0 <5=LH5, 8;8 1>;LH5, ?@02>9 AB>@>=K. 5@025=AB20 <>6=> C?>@O4>G5==> @0A?>;>68BL 2 2845 ?>A;54>20B5;L=>AB8 =5@025=AB2: </p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%"><i><b>(z+1)<sup>n</sup> < z<sup>n</sup> + z<sup>n</sup> ;&nbsp;(z+1)<sup>n</sup> < z<sup>n</sup> + (z-1)<sup>n</sup>  & ; &nbsp;(z + 1)<sup>n</sup> > 1<sup>n</sup> + 1<sup>n</sup></b></i></td> <td width="10%">&nbsp;&nbsp;&nbsp;&nbsp;(16)</td> </tr> </table> <p>! 2>7@0AB0=85< ?>:070B5;O AB5?5=8 <b><i>n > 2</i></b> G8A;> =5@025=AB2 >48=0:>2>3> A<KA;0 ?@02>9 AB>@>=K ?>A;54>20B5;L=>AB8 (16) C25;8G8205BAO, 0 G8A;> =5@025=AB2 5Q ;52>9 AB>@>=K >48=0:>2>3>, => ?@>B82>?>;>6=>3> A<KA;0, C<5=LH05BAO 4> =C;O. 0?@8<5@, ?@8 ?>:070B5;5 AB5?5=8 <b><i>n = k</i></b> ;520O AB>@>=0 =5@025=AB2 ?>A;54>20B5;L=>AB8 (16) 8AG5705B, 0 2 >AB02H59AO 5Q ?@02>9 G0AB8 2A5 =5@025=AB20 8<5NB >48=0:>2K9 A<KA;: </p> <table border="0" align="center"> <tr> <td width="10%">&nbsp;</td> <td width="80%"><b><i>(z + 1)<sup>k</sup> > z<sup>k</sup> + z<sup>k</sup>; &nbsp;(z + 1)<sup>k</sup> &nbsp;> z<sup>k</sup> + (z  1)<sup>k</sup>; &nbsp; & ; &nbsp;(z + 1)<sup>k</sup> > 1<sup>k</sup> + 1<sup>k</sup> </i></b></td> <td width="10%">&nbsp;&nbsp;&nbsp;&nbsp;(17)</td> </tr> </table> <p>@8 40;L=59H5< 2>7@0AB0=88 ?>:070B5;O AB5?5=8 <b><i>n > k</i></b> 4> 15A:>=5G=>AB8 =8 >4=> 87 =5@025=AB2 ?>A;54>20B5;L=>AB8 (17) =5 87<5=O5B A2>53> A<KA;0 8 =5 >1@0I05BAO 2 @025=AB2>.  ?>A;54>20B5;L=>AB8 (16) =5@025=AB2>, >1@07>20==>5 87 ?@>872>;L=> 27OB>9 B@>9:8 F5;KE ?>;>68B5;L=KE G8A5; <b><i>(z + 1, x, y)</i></b>, <>65B =0E>48BLAO 2 5Q ?@02>9 G0AB8 2 2845 <b><i>(z + 1) <sup>n</sup>&nbsp;> x <sup>n</sup> + y <sup>n</sup></i></b> 8;8 =0E>48BLAO 2 5Q ;52>9 G0AB8 2 2845 <b><i>(z + 1) <sup>n</sup>&nbsp;< x <sup>n</sup> + y <sup>n</sup></i></b> . </p> <p> ;N1>< A;CG05 B@>9:0 F5;KE ?>;>68B5;L=KE G8A5; <b><i>(z + 1, x, y)</i></b> ?@8 <b><i>n > 2</i></b> , <b><i>z + 1 > x</i></b>, <b><i>z + 1 > y</i></b> 2 ?>A;54>20B5;L=>AB8 (16) ?@54AB02;O5B A>1>9 =5@025=AB2> 8 =5 <>65B ?@54AB02;OBL A>1>9 @025=AB20, B. 5. =5 <>65B ?@54AB02;OBL A>1>9 @5H5=8O C@02=5=8O 1>;LH>9 B5>@5<K $5@<0. </p> <p>53:> 8 ?@>AB> ?>=OBL ?@>8AE>645=85 ?>A;54>20B5;L=>AB8 AB5?5==KE =5@025=AB2 (16), 2 :>B>@>9 ?>A;54=55 =5@025=AB2> ;52>9 AB>@>=K 8 ?5@2>5 =5@025=AB2> ?@02>9 AB>@>=K O2;ONBAO =5@025=AB20<8 ?@>B82>?>;>6=>3> A<KA;0. 0>1>@>B, =5;53:> 8 =5?@>AB> H:>;L=8:0<, AB0@H5:;0AA=8:C 8 AB0@H5:;0AA=8F5, ?>=OBL, :0:8< >1@07>< 87 ?>A;54>20B5;L=>AB8 =5@025=AB2 (16) >1@07C5BAO ?>A;54>20B5;L=>ABL =5@025=AB2 (17), 2 :>B>@>9 2A5 =5@025=AB20 >48=0:>2>3> A<KA;0.</p> <p> ?>A;54>20B5;L=>AB8 (16) C25;8G5=85 F5;>9 AB5?5=8 =5@025=AB2 =0 1 548=8FC >1@0I05B ?>A;54=55 =5@025=AB2> ;52>9 AB>@>=K 2 ?5@2>5 =5@025=AB2> ?@>B82>?>;>6=>3> A<KA;0 ?@02>9 AB>@>=K. "0:8< >1@07><, :>;8G5AB2> =5@025=AB2 452>9 AB>@>=K ?>A;54>20B5;L=>AB8 C<5=LH05BAO, 0 :>;8G5AB2> =5@025=AB2 ?@02>9 AB>@>=K C25;8G8205BAO. 564C ?>A;54=8< 8 ?5@2K< AB5?5==K<8 =5@025=AB20<8 ?@>B82>?>;>6=>3> A<KA;0 2 >1O70B5;L=>< ?>@O4:5 =0E>48BAO AB5?5==>5 @025=AB2>. 3> AB5?5=L =5 <>65B 1KBL F5;K< G8A;><, B0: :0: <564C 42C<O ?>A;54>20B5;L=K<8 =0BC@0;L=K<8 G8A;0<8 =0E>4OBAO B>;L:> =5F5;K5 G8A;0. !B5?5==>5 @025=AB2> =5F5;>9 AB5?5=8, ?> CA;>28N B5>@5<K, =5 <>65B AG8B0BLAO @5H5=85< C@02=5=8O (1).</p> <p>A;8 2 ?>A;54>20B5;L=>AB8 (16) ?@>4>;60BL C25;8G5=85 AB5?5=8 =0 1 548=8FC, B> ?>A;54=55 =5@025=AB2> 5Q ;52>9 AB>@>=K >1@0B8BAO 2 ?5@2>5 =5@025=AB2> ?@>B82>?>;>6=>3> A<KA;0 ?@02>9 AB>@>=K.  @57C;LB0B5 =5 >AB0=5BAO =8 >4=>3> =5@025=AB20 ;52>9 AB>@>=K 8 >AB0=CBAO B>;L:> =5@025=AB20 ?@02>9 AB>@>=K, :>B>@K5 ?@54AB02OB A>1>9 ?>A;54>20B5;L=>ABL CA8;820NI8EAO AB5?5==KE =5@025=AB2 (17). 0;L=59H55 C25;8G5=85 8E F5;>9 AB5?5=8 =0 1 548=8FC ;8HL CA8;8205B 5Q AB5?5==K5 =5@025=AB20 8 :0B53>@8G5A:8 8A:;NG05B 2>7<>6=>ABL ?>O2;5=8O @025=AB20 2 F5;>9 AB5?5=8.</p> <p>!;54>20B5;L=>, 2>>1I5, =8:0:CN F5;CN AB5?5=L =0BC@0;L=>3> G8A;0 (z+1) ?>A;54>20B5;L=>AB8 AB5?5==KE =5@025=AB2 (17) =52>7<>6=> @07;>68BL =0 425 F5;KE AB5?5=8 A B5< 65 ?>:070B5;5<. >MB><C C@02=5=85 (1) =5 8<55B @5H5=89 =0 15A:>=5G=>< <=>65AB25 =0BC@0;L=KE G8A5;, GB> 8 B@51>20;>AL 4>:070BL.</p> <p>!;54>20B5;L=>, 1>;LH0O B5>@5<0 $5@<0 4>:070=0 2> 2A59 2A5>1I=>AB8: </p> <ol>&nbsp;- 2 @0745;5 ) 4;O 2A5E B@>5: <b><i>(z, x, y)</i></b> ?8D03>@>2KE G8A5; (>B:@KB>5 $5@<0 ?>8AB8=5 GC45A=>5 4>:070B5;LAB2>), </ol> <ol>&nbsp;- 2 @0745;5 ) 4;O 2A5E G;5=>2 A5<59AB20 ;N1>9 B@>9:8 <b><i>(z, x, y)</i></b> ?8D03>@>2KE G8A5;, </ol> <ol>&nbsp;- 2 @0745;5 !) 4;O 2A5E B@>5: G8A5; <b><i>(z, x, y)</i></b>, =5 1>;LH8E G8A;0 <b><i>z</i></b>, </ol> <ol>&nbsp;- 2 @0745;5 D) 4;O 2A5E B@>5: G8A5; <b><i>(z, x, y)</i></b> =0BC@0;L=>3> @O40 G8A5;. </ol> <table border="0" align="right"> <tr> <td><p><i>7<5=5=8O 2=5A5=K 05.09.2010 3.</i></p></td> </tr> </table> &nbsp; &nbsp; &nbsp; &nbsp; <table border="0" align="right"> <tr> <td>&nbsp;</td> </tr> </table> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<h3 align="center"><i>0:85 B5>@5<K <>6=> 8 :0:85 =5;L7O 4>:070BL >B ?@>B82=>3></i></h3> &nbsp; <p> B>;:>2>< A;>20@5 <0B5<0B8G5A:8E B5@<8=>2 40=> >?@545;5=85 4>:070B5;LAB2C >B ?@>B82=>3> B5>@5<K, ?@>B82>?>;>6=>9 >1@0B=>9 B5>@5<5.</p> <p><i>«>:070B5;LAB2> >B ?@>B82=>3>  <5B>4 4>:070B5;LAB20 B5>@5<K (?@54;>65=8O), A>AB>OI89 2 B><, GB> 4>:07K20NB =5 A0<C B5>@5<C, 0 59 @02=>A8;L=CN (M:2820;5=B=CN), ?@>B82>?>;>6=CN >1@0B=>9 (>1@0B=CN ?@>B82>?>;>6=>9) B5>@5<C. >:070B5;LAB2> >B ?@>B82=>3> 8A?>;L7CNB 2AO:89 @07, :>340 ?@O<CN B5>@5<C 4>:070BL B@C4=>, 0 ?@>B82>?>;>6=CN >1@0B=>9 ;53G5. @8 4>:070B5;LAB25 >B ?@>B82=>3> 70:;NG5=85 B5>@5<K 70<5=O5BAO 5Q >B@8F0=85<, 8 ?CBQ< @0AAC645=8O ?@8E>4OB : >B@8F0=8N CA;>28O, B.5. : ?@>B82>@5G8N, : ?@>B82=><C (?@>B82>?>;>6=><C B><C, GB> 40=>; MB> ?@82545=85 : 01AC@4C 8 4>:07K205B B5>@5<C».</i></p> <p><i>>:070B5;LAB2> >B ?@>B82=>3> >G5=L G0AB> ?@8<5=O5BAO 2 <0B5<0B8:5. >:070B5;LAB2> >B ?@>B82=>3> >A=>20=> =0 70:>=5 8A:;NGQ==>3> B@5BL53>, 70:;NG0NI53>AO 2 B><, GB> 87 42CE 2KA:07K20=89 (CB25@645=89)  8  (>B@8F0=85 ) >4=> 87 =8E 8AB8==>, 0 4@C3>5 ;>6=>».</i> /">;:>2K9 A;>20@L <0B5<0B8G5A:8E B5@<8=>2: >A>185 4;O CG8B5;59/. . 0=BC@>2 [8 4@.]; ?>4 @54. . . 8B:8=0.- .: @>A25I5=85, 1965.- 539 A.: 8;.-C.112/. </p> <p>5 ;CGH5 1K;> 1K >B:@KB> 70O28BL > B><, GB> <5B>4 4>:070B5;LAB20 >B ?@>B82=>3> =5 O2;O5BAO <0B5<0B8G5A:8< <5B>4><, E>BO 8 8A?>;L7C5BAO 2 <0B5<0B8:5, GB> >= O2;O5BAO ;>38G5A:8< <5B>4>< 8 ?@8=04;568B ;>38:5. >?CAB8<> ;8 CB25@640BL, GB> 4>:070B5;LAB2> >B ?@>B82=>3> «8A?>;L7CNB 2AO:89 @07, :>340 ?@O<CN B5>@5<C 4>:070BL B@C4=>», :>340 =0 A0<>< 45;5 53> 8A?>;L7CNB B>340, 8 B>;L:> B>340, :>340 5<C =5B 70<5=K.</p> <p>0A;C68205B >A>1>3> 2=8<0=8O 8 E0@0:B5@8AB8:0 >B=>H5=8O 4@C3 : 4@C3C ?@O<>9 8 >1@0B=>9 59 B5>@5<. <i>«1@0B=0O B5>@5<0 4;O 40==>9 B5>@5<K (8;8 : 40==>9 B5>@5<5) - B5>@5<0, 2 :>B>@>9 CA;>285< O2;O5BAO 70:;NG5=85, 0 70:;NG5=85<  CA;>285 40==>9 B5>@5<K. 0==0O B5>@5<0 ?> >B=>H5=8N : >1@0B=>9 B5>@5<5 =07K205BAO ?@O<>9 B5>@5<>9 (8AE>4=>9).  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"@51C5BAO 4>:070BL, :0:>5 87 =8E O2;O5BAO ;>6=K<.</p> <p>>:07K205< ?>A@54AB2>< ?@028;L=>3> <b>;>38G5A:>3></b> @0AAC645=8O 8 >1=0@C68205<, GB> 53> @57C;LB0B>< O2;O5BAO ;>6=>5, 01AC@4=>5 70:;NG5=85. @8G8=>9 ;>6=>3> ;>38G5A:>3> 70:;NG5=8O O2;O5BAO ?@>B82>@5G82>5 ;>38G5A:>5 CA;>285 B5>@5<K, 70:;NG0NI55 2 A515 425 G0AB8  ;>6=CN 8 8AB8==CN. >6=>9 G0ABLN <>65B 1KBL B>;L:> CB25@645=85 <b>87 <i></i> =5 A;54C5B <i></i></b>, 2 :>B>@>< <b><i></i></b> 1K;> ?@8=OB> 157 4>:070B5;LAB20. <5==> MB8< >=> >B;8G05BAO >B <b><i></i></b> CB25@645=8O <b>87 <i></i> A;54C5B <i></i></b>, :>B>@>5 4>:070=> 4>:070B5;LAB2>< ?@O<>9 B5>@5<K.</p> <p>!;54>20B5;L=>, 8AB8==K< O2;O5BAO CB25@645=85: <b>87 <i></i> A;54C5B <i></i></b>, GB> 8 B@51>20;>AL 4>:070BL.</p> <p><i><b>K2>4</b>: ;>38G5A:8< <5B>4>< >B ?@>B82=>3> 4>:07K205BAO B>;L:> B0 >1@0B=0O B5>@5<0, :>B>@0O 8<55B 4>:070==CN <0B5<0B8G5A:8< <5B>4>< ?@O<CN B5>@5<C 8 :>B>@CN <0B5<0B8G5A:8< <5B>4>< 4>:070BL =52>7<>6=>.</i></p> <p>>;CG5==K9 2K2>4 ?@8>1@5B05B 8A:;NG8B5;L=>5 ?> 206=>AB8 7=0G5=85 2 >B=>H5=88 : <5B>4C 4>:070B5;LAB20 >B ?@>B82=>3> 25;8:>9 B5>@5<K $5@<0. >402;ONI55 1>;LH8=AB2> ?>?KB>: 5Q 4>:070BL 8<55B 2 A2>59 >A=>25 =5 >1KG=K9 <0B5<0B8G5A:89 <5B>4, 0 ;>38G5A:89 <5B>4 4>:070B5;LAB20 >B ?@>B82=>3>. >:070B5;LAB2> 1>;LH>9 B5>@5<K $5@<0 #09;A0 =5 O2;O5BAO 8A:;NG5=85<. </p> <p><8B@89 1@0@>2 2 AB0BL5 <a href="http://www.polit.ru/science/2006/12/28/abrarov.html" title="«"5>@5<0 $5@<0: D5=><5= 4>:070B5;LAB2 #09;A0»">«"5>@5<0 $5@<0: D5=><5= 4>:070B5;LAB2 #09;A0»</a> >?C1;8:>20; :><<5=B0@89 : 4>:070B5;LAB2C 1>;LH>9 B5>@5<K $5@<0 #09;A><. > 1@0@>2C, #09;A 4>:07K205B 1>;LHCN B5>@5<C $5@<0 A ?><>ILN 70<5G0B5;L=>9 =0E>4:8 =5<5F:>3> <0B5<0B8:0 5@E0@40 $@5O (@. 1944), A2O702H53> ?>B5=F80;L=>5 @5H5=85 C@02=5=8O $5@<0 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, A 4@C38<, A>25@H5==> =5?>E>68< =0 =53>, C@02=5=85<. -B> =>2>5 C@02=5=85 7040QBAO A?5F80;L=>9 :@82>9 (=0720==>9 M;;8?B8G5A:>9 :@82>9 $@5O). @820O $@5O 7040QBAO C@02=5=85< A>2A5< =5A;>6=>3> 2840: <br><b><i>y<sup>2</sup> + x&nbsp;(x - a<sup>n</sup>)&nbsp;(y + b<sup>n</sup>) = 0</i></b>. </p> <p><i>« 8<5==> $@59 A>?>AB028; 2AO:><C @5H5=8N <b>(a, b, c)</b> C@02=5=85 $5@<0, B> 5ABL G8A;0<, C4>2;5B2>@ONI8< A>>B=>H5=8N <b>a<sup>n</sup> + b<sup>n</sup> = c<sup>n</sup></b>, C:070==CN 2KH5 :@82CN.  MB>< A;CG05 >BAN40 A;54>20;0 1K 25;8:0O B5>@5<0 $5@<0».</i>( &8B0B0 ?>: 1@0@>2 . <a href="http://www.polit.ru/science/2006/12/28/abrarov.html" title="«"5>@5<0 $5@<0: D5=><5= 4>:070B5;LAB2 #09;A0»">«"5>@5<0 $5@<0: D5=><5= 4>:070B5;LAB2 #09;A0»</a>)</p> <p>@C38<8 A;>20<8, 5@E0@4 $@59 ?@54?>;>68;, GB> C@02=5=85 1>;LH>9 B5>@5<K $5@<0 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, 8<55B @5H5=8O 2 F5;KE ?>;>68B5;L=KE G8A;0E. -B8<8 65 @5H5=8O O2;ONBAO, ?> ?@54?>;>65=8N $@5O, @5H5=8O<8 53> C@02=5=8O <br><b><i>y<sup>2</sup> + x&nbsp;(x - a<sup>n</sup>)&nbsp;(y + b<sup>n</sup>) = 0</i></b>, :>B>@>5 7040QBAO 53> M;;8?B8G5A:>9 :@82>9.</p> <p>-=4@N #09;A ?@8=O; MBC 70<5G0B5;L=CN =0E>4:C $@5O 8 A 5Q ?><>ILN ?>A@54AB2>< <b><0B5<0B8G5A:>3></b> <5B>40 4>:070;, GB> MB>9 =0E>4:8, B> 5ABL M;;8?B8G5A:>9 :@82>9 $@5O, =5 ACI5AB2C5B. >MB><C =5 ACI5AB2C5B C@02=5=8O 8 53> @5H5=89, :>B>@K5 7040NBAO =5ACI5AB2CNI59 M;;8?B8G5A:>9 :@82>9, >MB><C #09;AC A;54>20;> 1K ?@8=OBL 2K2>4 > B><, GB> =5 ACI5AB2C5B C@02=5=8O 1>;LH>9 B5>@5<K $5@<0 8 A0<>9 B5>@5<K $5@<0. 4=0:> 8< ?@8=8<05BAO 1>;55 A:@><=>5 70:;NG5=85 B><, GB> C@02=5=85 1>;LH>9 B5>@5<K $5@<0 =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E.</p> <p>5>?@>25@68<K< D0:B>< <>65B O2;OBLAO B>, GB> #09;A>< ?@8=OB> ?@54?>;>65=85, ?@O<> ?@>B82>?>;>6=>5 ?> A<KA;C B><C, GB> CB25@6405BAO 1>;LH>9 B5>@5<>9 $5@<0. => >1O7K205B #09;A0 4>:07K20BL 1>;LHCN B5>@5<C $5@<0 <5B>4>< >B ?@>B82=>3>. >A;54C5< 8 <K 53> ?@8<5@C 8 ?>A<>B@8<, GB> 87 MB>3> ?@8<5@0 ?>;CG05BAO.</p> <p> 1>;LH>9 B5>@5<5 $5@<0 CB25@6405BAO, GB> C@02=5=85 , <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E. </p> <p>!>3;0A=> ;>38G5A:><C <5B>4C 4>:070B5;LAB20 >B ?@>B82=>3>, MB> CB25@645=85 A>E@0=O5BAO, ?@8=8<05BAO :0: 40==>5 157 4>:070B5;LAB20, 8 70B5< 4>?>;=O5BAO ?@>B82>?>;>6=K< ?> A<KA;C CB25@645=85<: C@02=5=85 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, 8<55B @5H5=8O 2 F5;KE ?>;>68B5;L=KE G8A;0E.</p> <p>@54?>;>65==>5 CB25@645=85 B0: 65 ?@8=8<05BAO :0: 40==>5, 157 4>:070B5;LAB20. 10 CB25@645=8O, @0AA<0B@8205<K5 A B>G:8 7@5=8O >A=>2=KE 70:>=>2 ;>38:8, O2;ONBAO >48=0:>2> 4>?CAB8<K<8, @02=>?@02=K<8 8 >48=0:>2> 2>7<>6=K<8. >A@54AB2>< ?@028;L=>3> @0AAC645=8O B@51C5BAO CAB0=>28BL, 8<5==> :0:>5 87 =8E O2;O5BAO ;>6=K<, GB>1K 70B5< CAB0=>28BL, GB> 4@C3>5 CB25@645=85 O2;O5BAO 8AB8==K<.</p> <p>@028;L=>5 @0AAC645=85 7025@H05BAO ;>6=K<, 01AC@4=K< 70:;NG5=85<, ;>38G5A:>9 ?@8G8=>9 :>B>@>3> <>65B 1KBL B>;L:> ?@>B82>@5G82>5 CA;>285 4>:07K205<>9 B5>@5<K, 70:;NG0NI55 2 A515 425 G0AB8 ?@O<> ?@>B82>?>;>6=>3> A<KA;0. =8 8 O28;8AL ;>38G5A:>9 ?@8G8=>9 01AC@4=>3> 70:;NG5=8O, @57C;LB0B0 4>:070B5;LAB20 >B ?@>B82=>3>.</p> <p>4=0:> 2 E>45 ;>38G5A:8 ?@028;L=>3> @0AAC645=8O =5 1K;> >1=0@C65=> =8 >4=>3> ?@87=0:0, ?> :>B>@><C <>6=> 1K;> 1K CAB0=>28BL, :0:>5 8<5==> CB25@645=85 O2;O5BAO ;>6=K<. < <>65B 1KBL CB25@645=85: C@02=5=85 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E. 0 MB>< 65 >A=>20=88 8< <>65B 1KBL CB25@645=85: C@02=5=85 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E. </p> <p> 8B>35 @0AAC645=8O 2K2>4 <>65B 1KBL B>;L:> >48=: <b>1>;LHCN B5>@5<C $5@<0 <5B>4>< >B ?@>B82=>3> 4>:070BL =52>7<>6=></b>.</p> <p> 8B>35 @0AAC645=8O 2K2>4 <>65B 1KBL B>;L:> >48=: <b>1>;LHCN B5>@5<C $5@<0 <5B>4>< >B ?@>B82=>3> 4>:070BL =52>7<>6=></b>.</p> <p>K;> 1K A>2A5< 4@C3>5 45;>, 5A;8 1K 1>;LH0O B5>@5<0 $5@<0 1K;0 >1@0B=>9 B5>@5<>9, :>B>@0O 8<55B ?@O<CN B5>@5<C, 4>:070==CN >1KG=K< <0B5<0B8G5A:8< <5B>4><.  MB>< A;CG05 5Q <>6=> 1K;> 4>:070BL >B ?@>B82=>3>.  B0: :0: >=0 O2;O5BAO ?@O<>9 B5>@5<>9, B> 5Q 4>:070B5;LAB2> 4>;6=> 8<5BL 2 A2>59 >A=>25 =5 ;>38G5A:89 <5B>4 4>:070B5;LAB20 >B ?@>B82=>3>, 0 >1KG=K9 <0B5<0B8G5A:89 <5B>4.</p> <p>> A;>20< . 1@0@>20, A0<K9 8725AB=K9 87 A>2@5<5==KE @>AA89A:8E <0B5<0B8:>2 0:045<8: . . @=>;L4 =0 4>:070B5;LAB2> #09;A0 >B@5038@>20; «0:B82=> A:5?B8G5A:8». :045<8: 70O28;: «MB> =5 =0AB>OI0O <0B5<0B8:0  =0AB>OI0O <0B5<0B8:0 35><5B@8G=0 8 A8;L=0 A2O7O<8 A D878:>9».( &8B0B0 ?>: 1@0@>2 . <a href="http://www.polit.ru/science/2006/12/28/abrarov.html" title="«"5>@5<0 $5@<0: D5=><5= 4>:070B5;LAB2 #09;A0»">«"5>@5<0 $5@<0: D5=><5= 4>:070B5;LAB2 #09;A0»</a>). 0O2;5=85 0:045<8:0 2K@0605B A0<CN ACI=>ABL =5<0B5<0B8G5A:>3> 4>:070B5;LAB20 #09;A0 1>;LH>9 B5>@5<K $5@<0. </p> <p>5B>4>< >B ?@>B82=>3> =52>7<>6=> 4>:070BL =8 B>3>, GB> C@02=5=85 1>;LH>9 B5>@5<K $5@<0 =5 8<55B @5H5=89, =8 B>3>, GB> >=> 8<55B @5H5=8O. H81:0 #09;A0 =5 <0B5<0B8G5A:0O, 0 ;>38G5A:0O - 8A?>;L7>20=85 4>:070B5;LAB20 >B ?@>B82=>3> B0<, 345 53> 8A?>;L7>20=85 =5 8<55B A<KA;0 8 1>;LH>9 B5>@5<K $5@<0 =5 4>:07K205B. </p> <p>5 4>:07K205BAO 1>;LH0O B5>@5<0 $5@<0 8 A ?><>ILN >1KG=>3> <0B5<0B8G5A:>3> <5B>40, 5A;8 2 =59 40=>: C@02=5=85 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E, 8 5A;8 2 =59 B@51C5BAO 4>:070BL: C@02=5=85 <b><i>x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup></i></b>, 345 <b><i>n > 2</i></b>, =5 8<55B @5H5=89 2 F5;KE ?>;>68B5;L=KE G8A;0E.  B0:>9 D>@<5 8<55BAO =5 B5>@5<0, 0 B02B>;>38O, ;8HQ==0O A<KA;0.</p> <table border="0" align="right" width="50%"> <tr> <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td> <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td> <td><p align="right"><i>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;>?>;=5=> 27.07.2008 3.</i></p></td> </tr> </table> <table border="0" summary=""> <tr> <td>&nbsp;</td> <td>&nbsp;</td> <td>&nbsp;</td> </tr> </table> <p><b>@8<5G0=85.</b> &nbsp;>Q 4>:070B5;LAB2> "$ >1AC640;>AL =0 >4=>< 87 <a href="http://www.diary.ru/~eek/p70769218.htm#form">D>@C<>2</a>. 48= 87 CG0AB=8:>2 Trotil, A?5F80;8AB 2 B5>@88 G8A5;, A45;0; A;54CNI55 02B>@8B5B=>5 70O2;5=85 ?>4 =0720=85<: "@0B:89 ?5@5A:07 B>3>, GB> A45;0; 8@3>@>4A:89". @82>6C 53> 4>A;>2=>:</p> <p>"<b><i>.</i></b> <i>= 4>:070;, GB> 5A;8 <b><i>z<sup>2</sup> = x<sup>2</sup> + y <sup></sup></i></b>, B> &nbsp;<b><i>z<sup>n</sup> > x<sup>n</sup> + y <sup>n</sup></i></b>. -B> E>@>H> 8725AB=K9 8 2?>;=5 >G5284=K9 D0:B.</i></p> <p><b>.</b> <i>= 27O; 425 B@>9:8 - ?8D03>@>2C 8 =5 ?8D03>@>2C 8 ?>:070; ?@>ABK< ?5@51>@><, GB> 4;O :>=:@5B=>3>, >?@545;Q==>3> A5<59AB20 B@>5: (78 8 210 HBC:) "$ 2K?>;=O5BAO (8 B>;L:> 4;O =53>).</i></p> <p><b>!.</b> <i> 70B5< 02B>@>< >?CI5= B>B D0:B, GB> 87 <b><</b> 2 ?>A;54CNI59 AB5?5=8 <>65B >:070BLAO <b>=</b> , 0 =5 B>;L:> <b>></b>. @>AB>9 :>=B@?@8<5@ - ?5@5E>4 <b><i>n = 1</i></b> 2 <b><i>n = 2</i></b> 2 ?8D03>@>2>9 B@>9:5.</i></p> <p><b>D.</b> <i>-B>B ?C=:B =8G53> ACI5AB25==>3> 2 4>:070B5;LAB2> "$ =5 2=>A8B. 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A;8 >= ?@8HQ;AO =5 ?> 2:CAC Trotil,, B> 5<C A;54>20;> 1K ?@54;>68BL 4@C3>9 <5B>4, G53> >= =5 45;05B. 5 ?@54;030O =8G53> 270<5=, >AC640BL "?@>AB>9 ?5@51>@", :>B>@K9 2 40==>< A;CG05 =570<5=8<, =5:>@@5:B=>.</p> <p><b>!.</b> =>N >?CI5=> = <564C < 8 < =0 >A=>20=88 B>3>, GB> 2 4>:070B5;LAB25 "$ @0AA<0B@8205BAO C@02=5=85 <b><i>z<sup>2</sup> = x<sup>2</sup> + y <sup></sup></i></b> (1), 2 :>B>@>< AB5?5=L <b><i>n > 2</i></b> - <b>F5;>5</b> ?>;>68B5;L=>5 G8A;>. 7 @025=AB20, =0E>4OI53>AO <564C =5@025=AB20<8 A;54C5B <b>>1O70B5;L=>5</b> @0AA<>B@5=85 C@02=5=8O (1) <b>?@8 =5F5;>< 7=0G5=88 AB5?5=8</b> <b><i>n > 2</i></b>. Trotil, AG8B0O <b>>1O70B5;L=K<</b> @0AA<>B@5=85 @025=AB20 <564C =5@025=AB20<8, D0:B8G5A:8 AG8B05B <b>=5>1E>48<K<</b> 2 4>:070B5;LAB25 "$ @0AA<>B@5=85 C@02=5=8O (1) ?@8 <b>=5F5;><</b> 7=0G5=88 AB5?5=8 <b><i>n > 2</i></b>. / MB> A45;0; 4;O A51O 8 >1=0@C68;, GB> C@02=5=85 (1) ?@8 <b>=5F5;><</b> 7=0G5=88 AB5?5=8 <b><i>n > 2</i></b> 8<55B @5H5=85< B@>9:C G8A5;: <b><i>z, (z-1), (z-1)</i></b> ?@8 =5F5;>< ?>:070B5;5 AB5?5=8:</p> <table border="0" align="center" width="20%"> <tr> <td>&nbsp;</td> <td align="center"><b><i>lq 2</i></b></td> <td>&nbsp;</td> </tr> <tr> <td align="right"><b><i>n &nbsp;= </i></b></td> <td align="center"><img src="line_black2.jpg" width="50" height="1" border="0" alt=""></td> <td>&nbsp;</td> </tr> <tr> <td>&nbsp;</td> <td align="center"><b><i>lq z &sdot; lq (z-1)</i></b></td> <td>&nbsp;</td> </tr> </table> <p>A;8 CG5ABL, GB> $5@<0 =0?8A0;: "=52>7<>6=> @07;>68BL - 2>>1I5, =8:0:CN AB5?5=L, 1>;LHCN :204@0B0, =0 425 AB5?5=8 A B5< 65 ?>:070B5;5<", B> A;54C5B 70:;NG8BL, GB> >= >H810;AO 8 53> B5>@5<0 =525@=0. -B>B @57C;LB0B 1K; ?>;CG5= <=>N 2 1979 3. 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