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65

The smallest squares containingk 65's :
6561 = 812,   26656569 = 51632,   24656536576 = 1570242,
16546565656009 = 40677472,   6512656565865024 = 807010322.

65 is the second integer which is the sum of 2 squares in just 2 ways : 65 = 12 + 82 = 42 + 72.

112 + 292 + 472 + 652 = 862.

652 = 16 + 26 + 26 + 46 = 13 + 43 + 43 + 163.

652 = 4225, 4 and 225 are squares.

652 = 4225, 42 + 22 + 22 + 52 = 72,
652 = 4225, 42 + 2 + 5 = 72.

The sum of the consecutive odd primes 3 + 5 + ... + 199 = 652.

65 is the sum of m squares for m = 2,3,...,51.

652 = (12 + 4)(292 + 4).

652 = 163 + 15 + 27.

652 is the 4th square which is the sum of 4 sixth powers : 16 + 26 + 26 + 46.

652 is the first square which is the sum of 5 fourth powers in 3 ways :
  [2,2,2,3,8], [2,4,4,6,7], [3,4,6,6,6].
652 is the first square which is the sum of 9 fifth powers in 2 ways.

The 3-by-3 magic square consisting of different squares with constant 652:

02252602
392482202
522362152

A cubic Polynomial : (X + 652)(X + 722)(X + 47042) = X3 + 47052X2 + 4563122X + 220147202.

658 = 318644812890625, 3186 + 44 + 81 + 289 + 0625 = 652.
    32 + 12 + 82 + 62 + 42 + 42 + 82 + 12 + 22 + 82 + 92 + 0622 + 52 = 652.

22k + 24k + 58k + 65k are squares for k = 1,2,3 (132, 932, 7032).
230k + 610k + 665k + 2720k are squares for k = 1,2,3 (652, 28752, 1437252).
430k + 740k + 1470k + 1585k are squares for k = 1,2,3 (652, 23252, 874252).
481k + 624k + 1404k + 1716k are squares for k = 1,2,3 (652, 23532, 904152).

Komachi Fractions : 652 = 3460275/819, (2/65)2 = 396/418275, (65/73)2 = 38025/47961.

Komachi equations:
652 = 12 * 345 + 6 + 7 + 8 * 9 = 12 / 3 + 4 + 5 + 6 * 78 * 9
  = 123 * 4 - 5 + 6 * 7 * 89 = - 1 - 2 * 3 + 4 * 5 + 6 * 78 * 9
  = - 1 + 2 + 3 + 4 + 5 + 6 * 78 * 9 = - 1 + 23 - 4 - 5 + 6 * 78 * 9
  = 98 + 76 * 54 + 3 + 2 * 10 = 9 - 8 - 76 + 5 * 43 * 2 * 10
  = 987 + 6 * 543 - 2 * 10 = - 9 - 8 * 7 - 6 * 5 + 432 * 10
  = - 9 * 8 + 7 - 6 * 5 + 432 * 10 = - 9 - 87 + 6 - 5 + 432 * 10,
652 = 12 - 22 + 32 - 42 - 52 * 62 - 72 + 82 * 92 = 122 + 32 - 42 * 52 + 672 + 82 - 92
  = 122 * 32 / 42 + 52 / 62 * 782 - 92 = - 12 + 22 + 342 - 52 + 62 + 72 * 82 - 92
  = - 122 / 32 * 42 - 52 + 672 - 82 + 92 = - 122 * 32 / 42 + 52 / 62 * 782 + 92
  = 92 * 82 - 72 - 62 * 52 - 42 + 32 - 22 + 12.

563 + 653 = 6712.

(652 - 1) = (32 - 1)(232 - 1),   (652 + 9) = (72 + 9)(82 + 9),
(652 - 7) = (32 - 7)(462 - 7) = (82 - 7)(92 - 7).

652 + 662 + 672 + ... + 2822 = 27252,
652 + 662 + 672 + ... + 9282 = 163322.

(1 + 2 + ... + 32)(33 + 34 + ... + 65) = 9242,
(1 + 2 + ... + 44)(45 + 46 + ... + 55)(56 + 57 + ... + 65) = 181502,
(1 + 2 + ... + 54)(55 + 56 + ... + 65) = 9902.

(12)(22 + 32 + 42 + 52)(62 + 72 + ... + 422)(432 + 442 + ... + 652) = 3063602,
(12 + 22)(32 + 42 + 52)(62 + 72 + ... + 542)(552 + 562 + ... + 652) = 7315002,
(12 + 32 + ... + 62)(72 + 82 + ... + 382)(392)(402 + 412 + ... + 652) = 138411002,
(12 + 32 + ... + 132)(142 + 152 + ... + 172)(182 + 192 + ... + 262)(272 + 282 + ... + 652) = 174807362.

(13 + 23 + ... + 233)(243 + 253 + ... + 513)(523 + 533 + ... + 653) = 6035374802,
(13 + 23 + ... + 323)(333 + 343 + ... + 653) = 10977122.

652 = 4225 appears in the decimal expressions of π and e:
  π = 3.14159•••4255••• (from the 1839th digit),
  (4225 is the tenth 4-digit square in the expression of π,)
  e = 2.71828•••4255••• (from the 1246th digit).


Page of Squares : First Upload February 9, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan