a^2+b^2+c^2=d^2（0<a<b<c<d） 恰好有44组正整数解

northwolves

A374805
a(n) is the smallest positive integer whose square can be represented as the sum of 3 distinct nonzero squares in exactly n ways, or -1 if no such number exists.

1, 7, 15, 23, 31, 21, 33, 39, 49, 45, 79, 57, 95, 103, 75, 69, 127, 87, 63, 151, 93, 167, 111, 123, 99, 187, 117, 105, 161, 241, 141, 135, 153, 247, 271, 283, 177, 183, 165, 275, 147, 171, 323, 219

若 $a^2+b^2+c^2=d^2（0<a<b<c<d）$ 恰好有44组正整数解，求最小d值。已知d<15000内无解，如何证明不存在这样的d呢？

1. a[n_] :=NestWhile[# + 1 &, 1,Length@Select[PowersRepresentations[#^2, 3, 2],0 < #[[1]] < #[[2]] < #[[3]] &] != n &];
   
2. Range[20] // Map[{#, a[#]} &] // Column

复制代码

{1,7}
{2,15}
{3,23}
{4,31}
{5,21}
{6,33}
{7,39}
{8,49}
{9,45}
{10,79}
{11,57}
{12,95}
{13,103}
{14,75}
{15,69}
{16,127}
{17,87}
{18,63}
{19,151}
{20,93}

northwolves

遍历发现，15000以内，有38个数字无解：
{38,{44,78,174,183,205,210,282,295,330,372,384,404,434,460,468,479,480,509,582,624,652,670,674,678,709,744,745,750,759,782,804,810,839,866,867,897,953,999}}

northwolves

突然发现，a^2+b^2+c^2=d^2（0<a<b<c<d） 若仅有唯一正整数解，则d=7 或d={9, 11, 13, 14, 17}的$2^k(k >=0)$倍.

{7,9,11,13,14,17,18,22,26,28,34,36,44,52,56,68,72,88,104,112,136,144,176,208,224,272,288,352,416,448,544,576,704,832,896,1088,1152,1408,1664,1792,2176,2304,2816,3328,3584,4352,4608,5632,6656,7168,8704,9216,11264,13312,14336,17408,18432}

northwolves

A004432
Numbers that are the sum of 3 distinct nonzero squares.

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 56, 59, 61, 62, 65, 66, 69, 70, 74, 75, 77, 78, 81, 83, 84, 86, 89, 90, 91, 93, 94, 98, 101, 104, 105, 106, 107, 109, 110, 113, 114, 115, 116, 117, 118, 120, 121, 122, 125, 126, 129, 131, 133
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OFFSET
1,1
COMMENTS
Numbers that can be written as a(n) = x^2 + y^2 + z^2 with 0 < x < y < z.
This is a subsequence (equal to the range) of A024803. As a set, it is the union of A025339 and A024804, subsequences of numbers having exactly one, resp. more than one, such representations. - M. F. Hasler, Jan 25 2013
Conjecture: a number n is a sum of 3 squares, but not a sum of 3 distinct nonzero squares (i.e., is in A004432 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 22, 25, 27, 33, 34, 37, 43, 51, 57, 58, 67, 73, 82, 85, 97, 99, 102, 123, 130, 163, 177, 187, 193, 267, 627, 697}. - Jeffrey Shallit, Jan 15 2017

wayne

如果是本原解,gcd(a,b,c,d)=1,那么是这个数列 https://oeis.org/A097266

wayne

我觉得 题目  要么 改成 统计 本原解$GCD(a,b,c)=1,0<d$的个数,  要么统计 $0<a<=b<=c , 0<d$ 非零解 的个数 比较合适.

northwolves

我是奇怪 A004432 为什么算到43就不继续了
Numbers that are the sum of 3 distinct nonzero squares.

northwolves

$324^2=x^2 + y^2 + z^2 with 0 < x < y < z$ 有18组解
{{4,112,304},{4,176,272},{24,84,312},{24,120,300},{24,168,276},{32,64,316},{32,196,256},{36,144,288},{64,92,304},{64,164,272},{64,188,256},{80,176,260},{80,220,224},{84,168,264},{92,176,256},{112,164,256},{128,196,224},{160,176,220}}

$1156^2=x^2 + y^2 + z^2 with 0 < x < y < z$ 有34组解
{{4,96,1152},{4,768,864},{68,336,1104},{96,612,976},{96,688,924},{112,324,1104},{112,576,996},{156,240,1120},{156,480,1040},{156,544,1008},{156,752,864},{176,336,1092},{176,672,924},{192,284,1104},{192,796,816},{228,256,1104},{228,576,976},{240,444,1040},{256,480,1020},{256,528,996},{284,768,816},{324,464,1008},{324,528,976},{324,752,816},{336,608,924},{336,768,796},{432,444,976},{432,544,924},{444,688,816},{464,612,864},{480,544,900},{508,576,864},{544,612,816},{576,644,768}}

wayne

northwolves 发表于 2025-10-4 07:51
遍历发现，15000以内，有38个数字无解：
{38,{44,78,174,183,205,210,282,295,330,372,384,404,434,460,468 ...

根据我这个数据, https://oeis.org/A097266.  应该是 无解了./