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71

The smallest squares containingk 71's :
17161 = 1312,   715716 = 8462,   3171717124 = 563182,
7121714171716 = 26686542,   1871717107115716 = 432633462.

71 is the first prime for which Legendre Symbol (a/71) = 1 for a = 1, 2, 3, 4, 5, 6.

712 = 5041, a zigzag square with different digits.

712 is the first square which is the sum of 3 cubes in 2 ways : 43 + 43 + 173 = 63 + 93 + 163,
712 is the second square which is the sum of 7 fourth powers in 2 ways :
    24 + 24 + 24 + 24 + 44 + 54 + 84 = 24 + 44 + 44 + 54 + 64 + 64 + 64.

712 = 7! + 1!. (The next is 415456).

A1 + B1 + C1, A2 + B2 + C2, A3 + B3 + C3 are squares for A = 34, B = 2134, C = 2873:
  341 + 21341 + 28731 = 712, 342 + 21342 + 28732 = 35792, 343 + 21343 + 28733 = 1828452.
See 19. The next primitive solution is A = 34813720, B = 42217144, C = 44343425.

A 3-by-3 magic square consisting of different squares with constant 712:

32262662
462512182
542422192

Komachi equations:
712 = 1 * 2 * 3 * 4 * 5 * 6 * 7 - 8 + 9 = - 1 + 2 + 3 * 4 * 5 / 6 * 7 * 8 * 9
  = - 1 + 2 + 3 * 45 * 6 * 7 * 8 / 9 = 9 - 8 + 7 * 6 * 5 * 4 * 3 * 2 */ 1
  = 9 * 8 * 7 / 6 * 5 * 4 * 3 + 2 - 1 = 98 / 7 * 6 * 5 * 4 * 3 + 2 - 1
  = 9 * 8 * 76 - 5 * 43 * 2 - 1 = 98 / 7 / 6 * 5 * 432 + 1,
712 = 12 - 22 * 32 / 42 * 562 / 72 + 82 * 92 = 122 - 32 + 42 * 52 + 672 - 82 + 92
  = - 12 * 22 + 32 * 42 * 52 / 62 * 72 + 82 + 92 = - 12 - 232 + 42 * 52 + 62 - 72 + 82 * 92
  = - 122 * 32 - 42 + 562 + 72 * 82 + 92 = - 12 + 232 + 42 + 52 + 672 + 82 - 92
  = 92 + 82 + 72 / 62 * 52 * 42 * 32 - 22 */ 12 = - 982 - 72 * 62 + 52 + 42 * 322 */ 12
  = - 92 + 82 + 72 + 652 + 42 / 32 * 212 = 92 * 82 + 72 + 62 - 542 / 32 - 22 + 102.

(712 - 1) = (22 - 1)(412 - 1) = (82 - 1)(92 - 1) = (22 - 1)(62 - 1)(72 - 1),
(712 + 8) = (32 + 8)(172 + 8) = (12 + 8)(32 + 8)(52 + 8).

(1 + 2 + ... + 7)(8 + 9 + ... + 19)(20 + 21 + ... + 71) = 32762,
(1 + 2 + ... + 7)(8 + 9 + ... + 27)(28 + 29 + ... + 71) = 46202,
(1 + 2 + ... + 13)(14 + 15 + ... + 64)(65 + 66 + ... + 71) = 92822,
(1 + 2 + ... + 33)(34 + 35 + ... + 54)(55 + 56 + ... + 71) = 235622.

(13 + 23 + 33)(43 + 53 + ... + 83)(93 + 103 + ... + 713) = 5443202.

718 = 645753531245761, 6 + 4575 + 35 + 312 + 45 + 7 + 61 = 712,
718 = 645753531245761, 6 + 4575 + 353 + 1 + 24 + 5 + 76 + 1 = 712,
718 = 645753531245761, 64 + 5 + 75 + 3 + 5 + 312 + 4576 + 1 = 712,
718 = 645753531245761, 645 + 75 + 3531 + 24 + 5 + 761 = 712.

712 = 5041 appears in the decimal expressions of π and e:
  π = 3.14159•••5041••• (from the 11493rd digit),
  e = 2.71828•••5041••• (from the 6582nd digit).

Page of Squares : First Upload February 16, 2004 ; Last Revised January 11, 2011
by Yoshio Mimura, Kobe, Japan