Encontrar todos los enteros soluciones a $$x^4 + y^4 + z^3 = 5$$ I don't know how to proceed, since it has a p-adic and real solution for all $p$. Creo que el único que es (2, 2, -3) y el trivial de los que vienen desde el este, pero no lo puedo confirmar.
Respuestas
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Stephan Aßmus
Puntos
16
Soluciones a $$ z^3 + 5 = k^4 + s^2 $$ razonablemente frecuentes
September 27
z: 3 n: 32 k: 2 sq: 4 sq_sq: 2 diff 0
z: 5 n: 130 k: 3 sq: 7 sq_sq: 2 diff 3
z: 15 n: 3380 k: 2 sq: 58 sq_sq: 7 diff 9
z: 20 n: 8005 k: 9 sq: 38 sq_sq: 6 diff 2
z: 68 n: 314437 k: 23 sq: 186 sq_sq: 13 diff 17
z: 83 n: 571792 k: 4 sq: 756 sq_sq: 27 diff 27
z: 93 n: 804362 k: 19 sq: 821 sq_sq: 28 diff 37
z: 101 n: 1030306 k: 3 sq: 1015 sq_sq: 31 diff 54
z: 168 n: 4741637 k: 31 sq: 1954 sq_sq: 44 diff 18
z: 405 n: 66430130 k: 73 sq: 6167 sq_sq: 78 diff 83
z: 431 n: 80062996 k: 94 sq: 1410 sq_sq: 37 diff 41
z: 488 n: 116214277 k: 103 sq: 1914 sq_sq: 43 diff 65
z: 684 n: 320013509 k: 25 sq: 17878 sq_sq: 133 diff 189
z: 765 n: 447697130 k: 103 sq: 18307 sq_sq: 135 diff 82
z: 848 n: 609800197 k: 9 sq: 24694 sq_sq: 157 diff 45
z: 1005 n: 1015075130 k: 79 sq: 31243 sq_sq: 176 diff 267
z: 1221 n: 1820316866 k: 11 sq: 42665 sq_sq: 206 diff 229
z: 1391 n: 2691419476 k: 126 sq: 49390 sq_sq: 222 diff 106
z: 1871 n: 6549699316 k: 130 sq: 79146 sq_sq: 281 diff 185
z: 2480 n: 15252992005 k: 247 sq: 107382 sq_sq: 327 diff 453
z: 2768 n: 21207928837 k: 247 sq: 132234 sq_sq: 363 diff 465
z: 2888 n: 24087491077 k: 39 sq: 155194 sq_sq: 393 diff 745
z: 3485 n: 42326109130 k: 409 sq: 119763 sq_sq: 346 diff 47
z: 4076 n: 67717750981 k: 103 sq: 260010 sq_sq: 509 diff 929
z: 5636 n: 179024699461 k: 625 sq: 162594 sq_sq: 403 diff 185
z: 5763 n: 191401729952 k: 466 sq: 379796 sq_sq: 616 diff 340
z: 5925 n: 208000828130 k: 509 sq: 375337 sq_sq: 612 diff 793
z: 7965 n: 505309357130 k: 691 sq: 526613 sq_sq: 725 diff 988
z: 12308 n: 1864500322117 k: 1167 sq: 98786 sq_sq: 314 diff 190
z: 14213 n: 2871159161602 k: 1269 sq: 527159 sq_sq: 726 diff 83
z: 14991 n: 3368928644276 k: 1318 sq: 592730 sq_sq: 769 diff 1369
z: 15615 n: 3807377733380 k: 1052 sq: 1607042 sq_sq: 1267 diff 1753
z: 17496 n: 5355700839941 k: 1423 sq: 1120430 sq_sq: 1058 diff 1066
z: 18063 n: 5893450576052 k: 686 sq: 2381594 sq_sq: 1543 diff 745
z: 19923 n: 7907955283472 k: 1292 sq: 2263076 sq_sq: 1504 diff 1060
z: 22901 n: 12010562298706 k: 571 sq: 3450255 sq_sq: 1857 diff 1806
z: 23445 n: 12886966846130 k: 1111 sq: 3370967 sq_sq: 1836 diff 71
z: 24693 n: 15056414740562 k: 667 sq: 3854671 sq_sq: 1963 diff 1302
z: 24995 n: 15615626874880 k: 464 sq: 3945792 sq_sq: 1986 diff 1596
z: 27663 n: 21168877523252 k: 1862 sq: 3024646 sq_sq: 1739 diff 525
z: 33180 n: 36528273432005 k: 1831 sq: 5028778 sq_sq: 2242 diff 2214
z: 35324 n: 44076756492229 k: 705 sq: 6620402 sq_sq: 2573 diff 73
z: 37388 n: 52263284795077 k: 2111 sq: 5692494 sq_sq: 2385 diff 4269
z: 38733 n: 58109000778842 k: 2203 sq: 5878381 sq_sq: 2424 diff 2605
z: 51293 n: 134950439050762 k: 2973 sq: 7538389 sq_sq: 2745 diff 3364
z: 53871 n: 156338201695316 k: 2602 sq: 10511890 sq_sq: 3242 diff 1326
z: 54344 n: 160492523139589 k: 2975 sq: 9064158 sq_sq: 3010 diff 4058
z: 61200 n: 229220928000005 k: 1367 sq: 15024278 sq_sq: 3876 diff 902
z: 61733 n: 235262196719842 k: 981 sq: 15308039 sq_sq: 3912 diff 4295
z: 63045 n: 250583197816130 k: 2803 sq: 13742407 sq_sq: 3707 diff 558
z: 68493 n: 321320597819162 k: 3911 sq: 9346411 sq_sq: 3057 diff 1162
z: 70995 n: 357835390324880 k: 4184 sq: 7168012 sq_sq: 2677 diff 1683
z: 80988 n: 531204838990277 k: 2887 sq: 21488054 sq_sq: 4635 diff 4829
z: 85020 n: 614558602008005 k: 3623 sq: 21030058 sq_sq: 4585 diff 7833
z: 86111 n: 638522048185636 k: 4820 sq: 9938694 sq_sq: 3152 diff 3590
z: 91551 n: 767342543357156 k: 4496 sq: 18940330 sq_sq: 4352 diff 426
z: 96200 n: 890277128000005 k: 3847 sq: 25908582 sq_sq: 5090 diff 482
jagy@phobeusjunior:
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z: 100221 n: 1006644663093866 k: 2573 sq: 31029275 sq_sq: 5570 diff 4375
z: 100980 n: 1029689061192005 k: 3617 sq: 29300722 sq_sq: 5413 diff 153
z: 100995 n: 1030147992574880 k: 5014 sq: 19952908 sq_sq: 4466 diff 7752
z: 121725 n: 1803596357953130 k: 1423 sq: 42420467 sq_sq: 6513 diff 1298
z: 125973 n: 1999090319542322 k: 5843 sq: 28870511 sq_sq: 5373 diff 1382
z: 126060 n: 2003235041016005 k: 4993 sq: 37171598 sq_sq: 6096 diff 10382
z: 130131 n: 2203648395038096 k: 5644 sq: 34480780 sq_sq: 5872 diff 396
z: 140480 n: 2772320878592005 k: 4617 sq: 48144778 sq_sq: 6938 diff 8934
z: 148191 n: 3254359196531876 k: 340 sq: 57046874 sq_sq: 7552 diff 14170
z: 160163 n: 4108531157450752 k: 3712 sq: 62599296 sq_sq: 7911 diff 15375
z: 161988 n: 4250583285982277 k: 7991 sq: 13152346 sq_sq: 3626 diff 4470
z: 163661 n: 4383647270173786 k: 1013 sq: 66201165 sq_sq: 8136 diff 6669
z: 165375 n: 4522822787109380 k: 5828 sq: 58044482 sq_sq: 7618 diff 10558
z: 165900 n: 4566034179000005 k: 5183 sq: 62003122 sq_sq: 7874 diff 3246
z: 176028 n: 5454378397973957 k: 7847 sq: 40778026 sq_sq: 6385 diff 9801
z: 199440 n: 7932987984384005 k: 6647 sq: 77336182 sq_sq: 8794 diff 1746
z: 200405 n: 8048698481430130 k: 5871 sq: 82828807 sq_sq: 9101 diff 606
z: 217475 n: 10285561814046880 k: 662 sq: 101416812 sq_sq: 10070 diff 11912
z: 220035 n: 10653082808542880 k: 94 sq: 103213772 sq_sq: 10159 diff 8491
z: 253023 n: 16198694022523172 k: 8032 sq: 109712186 sq_sq: 10474 diff 7510
z: 255080 n: 16596985896512005 k: 11343 sq: 6530798 sq_sq: 2555 diff 2773
z: 257363 n: 17046622002731152 k: 10524 sq: 69137876 sq_sq: 8314 diff 15280
z: 267188 n: 19074398313188677 k: 5871 sq: 133739714 sq_sq: 11564 diff 13618
z: 272288 n: 20187637882191877 k: 11849 sq: 21813174 sq_sq: 4670 diff 4274
z: 290516 n: 24519418580108101 k: 3145 sq: 156274074 sq_sq: 12500 diff 24074
z: 293693 n: 25332659542683562 k: 7269 sq: 150135829 sq_sq: 12252 diff 24325
z: 313919 n: 30935191351930564 k: 5260 sq: 173694258 sq_sq: 13179 diff 8217
z: 317283 n: 31940404348304192 k: 7564 sq: 169313176 sq_sq: 13012 diff 1032
timdev
Puntos
25910
Después de una cuidadosa investigación I se presentan algunos resultados que podrían ser útiles en la resolución final de este problema. Vamos a empezar con la ecuación original es decir, $$x^4+y^4+z^3=5$$ Es fácil ver que no hay solución de esta ecuación en la que el $x,y$ $z$ son todas positivas. En segundo lugar, si $(x,y,z)$ es una solución, así es $(-x,-y,z)$. A partir de estas observaciones podemos conseguir ese $z<0$. Vamos a reescribir la ecuación en términos de los valores positivos, es decir, $$x^4+y^4-z^3=5$$ donde $x,y$ $z$ son todas positivas. Esto es equivalente a $$x^4+y^4=5+z^3$$ A partir de esto, uno se $$x^4+y^4\equiv z^3\mod(5)$$ En primer lugar mostramos que $x\equiv 0\mod(5)$ $y\equiv 0\mod(5)$ fib $z\equiv 0\mod(5)$. Si $x\equiv 0\mod(5)$ $y\equiv 0\mod(5)$ $$x^4+y^4\equiv0\mod(5)\Rightarrow 5+z^3\equiv z^3\equiv0\mod(5)\Rightarrow z\equiv0\mod(5)$$ Ahora vamos a $z\equiv0\mod(5)$ $$5+z^3\equiv0\mod(5)\Rightarrow x^4+y^4\equiv0\mod(5)$$ Sin embargo, de cualquier $x\in\mathbb{Z}$ ha $x^4\equiv0\mod(5)$ o $x^4\equiv1\mod(5)$. Por lo tanto $$x^4+y^4\equiv0\mod(5)\Leftrightarrow x\equiv y\equiv0\mod(5)$$ Sin embargo, una inspección de nuestra ecuación modificada de los rendimientos que no podemos tener simultáneamente $x$ $y$ divisible por $5$ porque de lo contrario $z\equiv0\mod(5)$ y $$x^4+y^4\equiv0\mod(5^3)\Rightarrow 5+z^3\equiv0\mod(5^3)$$ que es imposible como $5+z^3\equiv 5\mod(5^3)$. Sabiendo que $x$ $y$ no puede ser simultáneamente divisible por $5$ $z$ no es divisible por $5$. La aplicación de Fermat poco teorema podemos reescribir la ecuación modificada como $$zx^4+zy^4\equiv1\mod(5)$$ Vamos a decir, sin pérdida de generalidad $y\equiv0\mod(5)$ $x^4\equiv1\mod(5)$ $$z\cdot 1+z\cdot 0\equiv1\mod(5)\Rightarrow z\equiv1\mod(5)$$
El otro exhaustiva caso serían $x^4\equiv1\mod(5)$$y^4\equiv1\mod(5)$, en cuyo caso $$z\cdot 1+z\cdot 1\equiv1\mod(5)\Rightarrow 2z\equiv1\mod(5)\Rightarrow z\equiv3\mod(5)$$ En este caso la inspección directa de $z=3$ rendimiento $x=\pm 2$$y=\pm 2$.
