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Sum of four squares (Posted on 2025-04-14)

There are infinitely many sets of positive integers A<B<C<D such that A^2+B^2, A^2+B^2+C^2, and A^2+B^2+C^2+D^2 are all squares. Find the value of A≤ 1500 which leads to the largest number of solutions.

No Solution Yet Submitted by K Sengupta

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A computer solution

| Comment 3 of 5 |

Algorithmic points which I hope made the program faster:

(1) Once A^2 or A^2+B^2 or A^2+B^2+C^2 has been calculated once, the total is put into a list or dictionary to use as a lookup table.

(2) I first found a complete answer for A^2+B^2, then searched only those (A,B) pairs combined with all possible C values.

Then limited the final search to only the (A,B,C) triplets which resolved to a square when checking all D values.

I thought about limiting the 2nd and 3rd searches to only those pair or triplets which had a fairly large count (I guess this would be a greedy algorithm), but ended up not doing so.

Results (I'll put code in a separate comment)

I started with maximum values for A,B,C,D as 1500, 1800, 2100, 2400 (ie not very large). This found A=216 as the value with the most solutions.

I increased maximum values for A,B,C,D as 1500, 2500, 3500, 4500. This found the most values for A when all conditions were met was A=288 (I found 121 such quadruplets).

Choosing larger values for maximum B,C,D might yield yet a different result for the "best A".

Here are the 121 A,B,C,D quadruplets found with A=288

[288, 330, 584, 1752],

[288, 384, 476, 507],

[288, 384, 476, 2145],

[288, 384, 476, 4368],

[288, 384, 504, 522],

[288, 384, 504, 697],

[288, 384, 504, 928],

[288, 384, 504, 1305],

[288, 384, 504, 1610],

[288, 384, 504, 2030],

[288, 384, 504, 2475],

[288, 384, 504, 3328],

[288, 384, 504, 4147],

[288, 384, 550, 1752],

[288, 384, 640, 840],

[288, 384, 640, 1122],

[288, 384, 640, 1155],

[288, 384, 640, 1500],

[288, 384, 640, 1920],

[288, 384, 640, 2436],

[288, 384, 640, 3150],

[288, 384, 640, 3960],

[288, 384, 693, 1124],

[288, 384, 728, 1635],

[288, 384, 900, 931],

[288, 384, 900, 1071],

[288, 384, 900, 1265],

[288, 384, 900, 1360],

[288, 384, 900, 1547],

[288, 384, 900, 1584],

[288, 384, 900, 2448],

[288, 384, 900, 2501],

[288, 384, 900, 2800],

[288, 384, 900, 2975],

[288, 384, 900, 3393],

[288, 384, 900, 3757],

[288, 384, 900, 4275],

[288, 384, 1152, 1265],

[288, 384, 1152, 1430],

[288, 384, 1152, 1664],

[288, 384, 1152, 1836],

[288, 384, 1152, 2135],

[288, 384, 1152, 2340],

[288, 384, 1152, 2560],

[288, 384, 1152, 2914],

[288, 384, 1152, 3211],

[288, 384, 1152, 3640],

[288, 384, 1152, 3960],

[288, 384, 1235, 3180],

[288, 384, 1400, 1554],

[288, 384, 1400, 2538],

[288, 384, 1400, 2775],

[288, 384, 1400, 3552],

[288, 384, 1768, 3435],

[288, 384, 1890, 2160],

[288, 384, 1890, 2600],

[288, 384, 1890, 4000],

[288, 384, 2376, 3232],

[288, 384, 2860, 3045],

[288, 384, 2860, 3705],

[288, 540, 759, 1080],

[288, 540, 759, 1300],

[288, 540, 759, 2000],

[288, 540, 759, 2340],

[288, 540, 759, 2728],

[288, 540, 759, 3740],

[288, 540, 759, 4004],

[288, 540, 816, 931],

[288, 540, 816, 1071],

[288, 540, 816, 1265],

[288, 540, 816, 1360],

[288, 540, 816, 1547],

[288, 540, 816, 1584],

[288, 540, 816, 2448],

[288, 540, 816, 2501],

[288, 540, 816, 2800],

[288, 540, 816, 2975],

[288, 540, 816, 3393],

[288, 540, 816, 3757],

[288, 540, 816, 4275],

[288, 540, 1309, 2244],

[288, 540, 1309, 3468],

[288, 540, 1680, 2384],

[288, 540, 1785, 2516],

[288, 540, 2565, 3516],

[288, 540, 2720, 2784],

[288, 616, 1056, 2355],

[288, 616, 1275, 2244],

[288, 616, 1275, 3468],

[288, 616, 1365, 1548],

[288, 616, 1365, 3660],

[288, 616, 1632, 1974],

[288, 616, 1632, 2415],

[288, 616, 1632, 2601],

[288, 616, 1632, 3315],

[288, 616, 1632, 3549],

[288, 616, 1632, 4455],

[288, 741, 1060, 1092],

[288, 741, 1060, 3180],

[288, 741, 1908, 2756],

[288, 741, 1908, 3960],

[288, 741, 1908, 4240],

[288, 840, 1184, 1554],

[288, 840, 1184, 2538],

[288, 840, 1184, 2775],

[288, 840, 1184, 3552],

[288, 840, 1225, 3816],

[288, 840, 1665, 2516],

[288, 1134, 1296, 2328],

[288, 1134, 1560, 2160],

[288, 1134, 1560, 2600],

[288, 1134, 1560, 4000],

[288, 1134, 2400, 3560],

[288, 1134, 2808, 4056],

[288, 1280, 2460, 2784],

[288, 1716, 1827, 3364],

[288, 1716, 2223, 3764],

[288, 1716, 2320, 2739],

[288, 1716, 2320, 3045],

[288, 1716, 2320, 3705],

[288, 1716, 3139, 4020]

Posted by Larry on 2025-04-16 12:55:55

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