\(a^4+b^4+1=c^4\)有整数解吗?

无心人

一组通项公式

\( a^2 + b^2 + 1 = c^2 \)
\( a = 10k + 2 \)
\( b = 10k^2 + 4k - 2 \)
\( c = 10k^2 + 4k + 3 \)

无心人

考虑另外一组，c - b = 5的，补上第一组
{8, 4, 9}
{18,30,35}
{28,76,81}
{38,142,147}

8        4
18    30     26
28    76     46
38  142     66

也是跟第一组一致的关系

无心人

另一组通项公式

\( a^2 + b^2 + 1 = c^2 \)
\( a = 10k - 2 \)
\( b = 10k^2 - 4k - 2 \)
\( c = 10k^2 - 4k + 3 \)

无心人

假设
\(x, x, y\)是一组解
那么设\(k\)是正整数
\( d = x + y \)
\(  a = 2dk \pm x \)
\(  b = 2dk^2 \pm 2xk - x \)
\(  c =  2dk^2 \pm 2xk + y \)
是一组通解

282842712474

无心人 发表于 2014-7-20 21:01
假设
\(x, x, y\)是一组解
那么设\(k\)是正整数

x x y就是最简单的pell方程，其特解通过连分数求得。

http://zh.wikipedia.org/zh/%E4%BD%A9%E5%B0%94%E6%96%B9%E7%A8%8B

282842712474

$a^2+b^2+1=c^2$通解应该是
$$\begin{aligned}
a&=-2 + 2 m + 4 n + 8 m n + 4 m^2 n + 2 n^2 + 4 m n^2 + 4 m^2 n^2\\
b&=2 + 4 m + 2 m^2 + 2 n + 4 m n + 4 m^2 n\\
c&=3 + 4 m + 2 m^2 + 4 n + 8 m n + 4 m^2 n + 2 n^2 + 4 m n^2 + 4 m^2 n^2
\end{aligned}$$
允许m、n为负数，计算结果为负时取绝对值，经检验没有办法获得解(72,144,161)，上面所列的其他解都可以获得...

补充：经检验，下面的解与上面的解产生的结果相同，只是下面的解形式相对简单一些，故同放在这里
$$\begin{aligned}
a&=-2 m + 4 m^2 n + 2 n^2 + 4 m n^2 + 4 m^2 n^2\\
b&=2 m^2 + 2 n + 4 m n + 4 m^2 n\\
c&=1 + 2 m^2 + 4 m^2 n + 2 n^2 + 4 m n^2 + 4 m^2 n^2
\end{aligned}$$

wayne

282842712474 发表于 2014-7-21 01:04
$a^2+b^2+1=c^2$通解应该是
$$\begin{aligned}
a&=2 + 2 m + 4 n + 8 m n + 4 m^2 n + 2 n^2 + 4 m n^2 + ...

你这个大概有90%的解不能获得。

1. {64,112,129},{72,144,161},{98,274,291},{104,172,201},{106,322,339},{128,268,297},{132,504,521},{136,148,201},{140,568,585},{142,254,291},{154,302,339},{162,438,467},{166,802,819},{174,882,899},{182,286,339},{186,582,611},{192,216,289},{200,1168,1185},{208,212,297},{208,1264,1281},{216,612,649},{220,820,849},{228,684,721},{234,1602,1619},{242,418,483},{242,526,579},{242,1714,1731},{244,1012,1041},{246,378,451},{252,456,521},{268,520,585},{268,2104,2121},{276,2232,2249},{278,562,627},{278,1318,1347},{288,756,809},{290,1118,1155},{294,342,451},{302,494,579},{302,1214,1251},{302,1558,1587},{302,2674,2691},{310,2818,2835},{318,342,467},{322,538,627},{336,1932,1961},{336,3312,3329},{338,746,819},{344,3472,3489},{348,1116,1169}, \[CenterEllipsis]2188191\[CenterEllipsis] ,{999978,26606418,26625203},{999978,133077174,133080931},{999978,346004814,346006259},{999978,1730027538,1730027827},{999978,2262343782,2262344003},{999978,5882094078,5882094163},{999978,7691969202,7691969267},{999978,29410470594,29410470611},{999978,38459846166,38459846179},{999978,99995600046,99995600051},

复制代码

无心人

wayne 发表于 2014-7-21 08:25
你这个大概有2/3的解不能获得。

测试下我的通解的覆盖率

无心人

差为17的那组，我现在怀疑是否跟存在
\( 2x^2 +  1 = y^2 \) 的复整数解有关

无心人

写了个haskell程序算100万的，要3分半多点，难道是笔记本太慢么