2元１次不定方程式( indefinite equation?)＜2/4＞「2022年度 大学入学共通テスト 本試験｜数学Ⅰ第4問」(配点20点)WolframAlphaで(合同式も)

つづき

https://qiita.com/mrrclb48z/items/af08059157cfbce8f0fe

(20220330)これも考察?を使っていません。
(20220404)これも考察?を使っていません。後半にWolframAlphaの合同式（mod）でやってみました。
オリジナル

https://waka-blog.com/?p=3665

# WolframAlphaで

https://www.wolframalpha.com/input?i=5%5E4*x-2%5E4*y%3D1&lang=ja

https://www.wolframalpha.com/input?i=5%5E4*x-2%5E4*y%3D1%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja


https://ja.wolframalpha.com/input?i=5%5E4x-2%5E4y%3D1+over+the+integers

https://www.wolframalpha.com/input?i=x+%3D+16+n+%2B+1+%2C+y+%3D+625+n+%2B+39+%2C+n%3D0&lang=ja

解 
### ア,イウ n = 0, x = 1, y = 39

https://www.wolframalpha.com/input?i=x+%3D+16+n+%2B+1+%2C+%EF%BD%98%3E%3D10%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja

結果
x = 16 n + 1 , n>=1 , n element Z

https://www.wolframalpha.com/input?i=x+%3D+16+n+%2B+1+%2C+y+%3D+625+n+%2B+39+%2C+n%3D1&lang=ja

解

### エオ,カキク n = 1, x = 17, y = 664




https://www.wolframalpha.com/input?i=625%5E2%3D5%5Ex&lang=ja

整数解
### ケ x = 8

https://www.wolframalpha.com/input?i=m%3D39%2C625%5E2%3D2%5E8*m%5E2%2B2%5Ex*m%2B1%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja

結果
### コ x = 5 , m = 39

https://www.wolframalpha.com/input?i=5%5E5*x-2%5E5*y%3D1%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja

結果
x = 32 n + 29 , y = 3125 n + 2832 , n element Z

https://www.wolframalpha.com/input?i=x+%3D+32+n+%2B+29+%2C+%EF%BD%98%3E%3D100%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja

結果
x = 32 n + 29 , n>=3 , n element Z

https://www.wolframalpha.com/input?i=x+%3D+32+n+%2B+29+%2C+y+%3D+3125+n+%2B+2832+%2C+n%3D3&lang=ja


### サシス,セソタチツ n = 3, x = 125, y = 12207


https://ja.wolframalpha.com/input?i=11%5E5x-2%5E5y%3D1+over+the+integers

https://www.wolframalpha.com/input?i=11%5E5*x-2%5E5*y%3D1%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja

結果
x = 32 n + 19 , y = 161051 n + 95624 , n element Z

https://www.wolframalpha.com/input?i=x+%3D+32+n+%2B+19+%2C+y+%3D+161051+n+%2B+95624++%2C+n%3D0&lang=ja

### テト,ナニヌネノ n = 0, x = 19, y = 95624



# wolfram alpha　合同式でやってみました。




roundup かっこ悪いですね。いい方法がありますか？



### (1) ア,イ,ウ

5^4*x-2^4*y=1


https://www.wolframalpha.com/input?i=5%5E4%EF%BC%8Ax%3D1+mod+2%5E4

General solution
x = 16 n + 1 and n element Z



https://www.wolframalpha.com/input?i=x+%3D+16+n+%2B+1%2C5%5E4*x-2%5E4*y%3D1%2Cn%3D0

Result

n = 0, x = 1, y = 39

### (1) エオ,カキク



https://www.wolframalpha.com/input?i=x+%3D+16+n+%2B+1+%2C+%EF%BD%98%3E%3D10%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3&lang=ja

整数解の例
n = 1 , x = 17



https://www.wolframalpha.com/input?i=5%5E4*x-2%5E4*y%3D1%2Cx+%3D+17

Solution
{625 x - 16 y = 1, x = 17}

### (2)ケ,コ

625^2=5^x

https://www.wolframalpha.com/input?i=625%5E2%3D5%5Ex&lang=ja

整数解
x = 8



https://www.wolframalpha.com/input?i=625%5E2%3D2%5E8*m%5E2%2B2%5Ex*m%2B1%2Cm%3D39+%E3%81%AE%E6%95%B4%E6%95%B0%E8%A7%A3

Integer solution
m = 39, x = 5



### (3)　サシス,セソタチツ 

5^5*x-2^5*y=1



https://www.wolframalpha.com/input?i=5%5E5%EF%BC%8Ax%3D1+mod+2%5E5

General solution
x = 32 n + 29 and n element Z


https://www.wolframalpha.com/input?i=32+*n+%2B+29%3E%3D100

Solution
n>=71/32

roundup(71/32)



https://www.wolframalpha.com/input?i=32*roundup%2871%2F32%29%2B29

Exact result
125



https://www.wolframalpha.com/input?i=5%5E5*x-2%5E5*y%3D1%2Cx+%3D+125

Solution
x = 125, y = 12207


### (4)　テト,ナニヌネノ  

11^5*x-2^5*y=1



https://www.wolframalpha.com/input?i=11%5E5*x%3D1+mod+2%5E5

General solution

x = 32 n + 19 and n element Z



https://www.wolframalpha.com/input?i=11%5E5*x-2%5E5*y%3D1%2Cx%3D19

Solution
x = 19, y = 95624




# 参考

https://en.wikipedia.org/wiki/Modular_arithmetic

https://ja.wikipedia.org/wiki/%E5%90%88%E5%90%8C%E7%AE%97%E8%A1%93

Congruence(合同:google翻訳より[UK]コングルアンス[US]カングルアンス)

https://mathworld.wolfram.com/Congruence.html

https://reference.wolfram.com/language/ref/Congruent.html


# つづく　＜3/4＞　へ