A380966 a(n) is an upper bound such that there exists an m X m magic square of n-th powers for all m >= a(n).

36, 52, 84, 140, 164, 196, 224, 252, 284, 312, 344, 372, 404, 436, 468, 500, 532, 564, 596, 632, 664, 696, 732, 764, 796, 832, 864, 900, 936, 968, 1004, 1036, 1072, 1108, 1144, 1180, 1212, 1248, 1284, 1320, 1356, 1392, 1428, 1464, 1500, 1536, 1572, 1608, 1644, 1680

Offset: 2

Author

Paolo Xausa, Feb 09 2025

Keywords

• nonn

Comments

See Rome and Yamagishi (2024), eq. (2.2).

In particular, an m X m magic square of squares is proved to exist for all m >= 36. Combined with previous results that show the existence of such squares for 4 <= m <= 64, it follows that an m X m magic square of squares exists for all m >= 4. The 3 x 3 case is still unsolved.

Links

• Brady Haran and Matt Parker, A Magic Square Breakthrough, YouTube Numberphile video, 2025.
• Nick Rome and Shuntaro Yamagishi, On the existence of magic squares of powers, arXiv:2406.09364v2 [math.NT], 2024.
• Index entries for sequences related to magic squares

Crossrefs

Cf. A364264.

Programs

Mathematica

A380966[n_] := 20 + 4*If[2 <= n <= 4, 2^n, Ceiling[n*(Log[n] + 4.20032)]];
Array[A380966, 50, 2]

Formula

a(n) = 4*2^n + 20, if 2 <= n <= 4;

a(n) = 4*ceiling(n*(log(n) + 4.20032)) + 20, if n >= 5. Cf. Rome and Yamagishi (2024), eq. (2.2).