一题 \(x^{131}+y^{137}+z^{139}=w^{149}\) 多解

Treenewbee

蔡家雄 发表于 2025-3-21 05:33
求 \(x^2 - 545*y^2= - 545\) 的正整数解，可以有吗？

\[x_n=\text{Round}\left[\frac{\sqrt{545}}{2}  \left(1961+84 \sqrt{545}\right)^n\right]\]
\[y_n=\text{Round}\left[\frac{1}{2}  \left(1961+84 \sqrt{545}\right)^n\right]\]

{45780,1961},
{179549160,7691041},
{704191759740,30164260841},
{2761839902151120,118304223327361},
{10831935392044932900,463989133725649001},
{42482847845760324682680,1819765264167772054561},
{166617718419136601360538060,7137118902076868272339241},
{653474649157005904775705588640,27991778514180213196342448641},
{2562927407376058739393715958108020,109783748195495894079186811230761},
{10051800638254253218896249211994065800,430571832430956382398357477304596001},{39423159540305773748452350015724767959580,1688702617010462736270463946801814285161},{154617621665278606387176897865423327943406960,6623091233343202420696377200999238321805441},{606410272748063153944734044975840276469274137540,25975762128469422883508455111855065896306654441},{2378340935100282024492640537218347698889165224024920,101876932444765843205917740252318367446076376912161},{9327852541053033351996982242236314699203029539351598700,399561303072609508584186493761137525268445653942841001},
......

ysr

蔡家雄 发表于 2025-3-21 01:31
求 \(x^2 - 545*y^2= 1\) 的正整数解，可以有吗？

x=1961  y=84
x=7691041  y=329448

5.480469是100000000内的解

Treenewbee

蔡家雄 发表于 2025-3-21 12:07
求 \(x^2 - 629*y^2= 1259\) 的通项公式，

求 \(x_{2n+1}\)= 628 , ...... 的通解公式，

\[x_n=\left\lceil \frac{1}{4} \left(313 \sqrt{629}+7850\right)^n \left(\left(8503-339 \sqrt{629}\right) (-1)^n+289 \sqrt{629}-7247\right)\right\rceil\]
\[y_n=\text{Round}\left[\frac{\left(313 \sqrt{629}+7850\right)^n \left(\left(8503-339 \sqrt{629}\right) (-1)^n+289 \sqrt{629}-7247\right)}{4 \sqrt{629}}\right]\]

{628,25},
{123638128,4929775},
{154672083128,6167179825},
{30475562417995628,1215140249609575},
{38125122079564885628,1520148167398609625},
{7511921441362867058073128,299519922556544636039375},
{9397461417642192662903408128,374701324822379615096289425},
{1851613531105375953394617018870628,73828666310002531235294566219175},
{2316380283629546866638324315712650628,92360130304870999449695443162219225},{456404222985391173451069614407788075388128,18198028106409856244672897541408942148975},{570964580744607565019312876214467221817613128,22765848703567912889396103862066687998399025},{112499077836477514463122965052138958123378802625628,4485631984345021604740468694325153997824165828775},{140737060649667477874445277267182911296433587843295628,5611554092474152162882941114882026199163308506828825},{27729897920911497756565985596556488441039942898160575583128,1105663436792468325844493231535288352901089089374139258575}

Treenewbee

蔡家雄 发表于 2025-3-21 20:43
请 Treenewbee ，

求 \(m=(2a+1)^2+16^2=(2b+1)^2+64^2\) 的正整数解，

{39065, 104585}

Treenewbee

蔡家雄 发表于 2025-3-21 23:02
求 \(x^2 - 629*y^2= - 1259\) 的通项公式，

求 \(x_{2n+1}\)= 7875 , ...... 的通解公式，

\[x_n=\left\lfloor \frac{1}{4} \left(313 \sqrt{629}+7850\right)^n \left(\left(339 \sqrt{629}-8503\right) (-1)^n+289 \sqrt{629}-7247\right)\right\rfloor\]

\[y_n=\left\lceil \frac{\left(313 \sqrt{629}+7850\right)^n \left(\left(339 \sqrt{629}-8503\right) (-1)^n+289 \sqrt{629}-7247\right)}{4 \sqrt{629}}\right\rceil\]

{{7875,314},
{9851725,392814},
{1941118617475,77397467814},
{2428351714961325,96824723645314},
{478466331903649977075,19077701996267795314},
{598564419077520419320925,23866326324982894757814},
{117937167107863344715398086675,4702462803215452782085982814},
{147540144855546843885103926930525,5882810823577686281994638730314},
{29070332556291569576158831911666946275,1159110065769502543609577471727030314},{36367170600524030661768535641792541790125,1450054051669285514937904739641480562814},{7165546329940973979473362522361104695260555875,285709042429744808810867035009697863465937814},{8964143954057509371327242818335671024329067899725,357423826096070284032804345572351741216345255314}}

ysr

蔡家雄 发表于 2025-3-23 03:10
设 \(m=493\)=(31^2+25)/2 是合数，

求 \(x^2 - 493*y^2= - 1\) 的正整数解和通项公式，

x=683982  y=30805

5.542969

ysr

蔡家雄 发表于 2025-3-23 03:15
设 \(m=697\)=(37^2+25)/2 是合数，

求 \(x^2 - 697*y^2= - 1\) 的正整数解和通项公式，

x=132  y=5
x=9200268  y=348485

5.457031

ysr

蔡家雄 发表于 2025-3-27 09:58
求 \(x^2 - 202*y^2= -1\) 的正整数解，，

x=3141  y=221

5.578125

ysr

蔡家雄 发表于 2025-3-27 09:59
求 \(x^2 - 226*y^2= -1\) 的正整数解，，

x=15  y=1
x=13545  y=901
x=12217575  y=812701

5.46875

ysr

蔡家雄 发表于 2025-3-27 10:00
求 \(x^2 - 298*y^2= -1\) 的正整数解，，

x=409557  y=23725

5.523438