The smallest squares containingk 96's :
196 = 142, 69696 = 2642, 9696996 = 31142,
10969629696 = 1047362, 2996969696311396 = 547445862.
The squares which begin with 96 and end in 96 are
9696996 = 31142, 96314596 = 98142, 96746896 = 98362, 960132196 = 309862,
961868196 = 310142,...
962 = 63 + 103 + 203.
962 = (32 - 1)(52 - 1)(72 - 1).
962 = 9216, a square with different digits.
962 = 9216, 9 + 21 + 6 = 62,
962 = 9216, 9 + 216 = 152.
962 = 322 + 642 + 642, 462 + 462 + 232 = 692.
69696 = 2642 and 9696996 = 31142 are squares.
A cubic polynomial : (X + 35)(X + 72)(X + 96) = X3 + 1252X2 + 80882X + 2419202.
(1 + 22 + 32) + (4 + 52 + 62) + (7 + 82 + 92) + ... + (94 + 952 + 962) = 4522.
102 + 96 = 142, 102 - 96 = 22.
Komachi equations:
962 = 12 * 3 * 45 * 6 - 7 * 8 * 9 = 12 / 3 * 4 * 56 / 7 * 8 * 9
= 98 / 7 * 654 + 3 * 2 * 10 = 9 + 8765 + 432 + 10,
962 = 982 + 72 + 62 - 52 - 42 + 32 - 212 = - 92 * 82 + 72 * 62 * 52 * 42 * 32 / 212.
(962 + 6) = (92 + 6)(102 + 6), (962 + 9) = (62 + 9)(142 + 9).
5312 = 382 + 392 + 402 + ... + 962.
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 96) = 27722,
(1 + 2 + ... + 8)(9 + 10 + ... + 68)(69 + 70 + ... + 96) = 138602,
(1 + 2 + ... + 13)(14 + 15 + ... + 85)(86 + 87 + ... + 96) = 180182,
(1 + 2 + ... + 27)(28 + 29 + ... + 71)(72 + 73 + ... + 96) = 415802,
(1 + 2 + ... + 45)(46 + 47 + ... + 50)(51 + 52 + ... + 96) = 289802.
962 = 9216 appears in the decimal expressions of π and e:
π = 3.14159•••9216••• (from the 990th digit),
(9216 is the third 4-digit square in the expression of π.)
e = 2.71828•••9216••• (from the 3708th digit).
966 = 782757789696, 7 + 82 + 75 + 77 + 8969 + 6 = 962.
968 = 7213895789838336,
7 + 2 + 1 + 3 + 8 + 9 + 5 + 789 + 8383 + 3 + 6 = 7 + 2 + 1 + 3 + 89 + 5 + 7 + 8983 + 83 + 36
= 7 + 2 + 1 + 38 + 9 + 57 + 8983 + 83 + 36 = 7 + 2 + 138 + 9 + 57 + 8983 + 8 + 3 + 3 + 6
= 7 + 21 + 3 + 8 + 95 + 7 + 8983 + 83 + 3 + 6 = 7 + 21 + 3 + 895 + 7898 + 383 + 3 + 6
= 7 + 21 + 3 + 8957 + 8 + 98 + 3 + 83 + 36 = 7 + 21 + 38 + 9 + 5 + 789 + 8 + 3 + 8336
= 7 + 213 + 8 + 9 + 578 + 9 + 8383 + 3 + 6 = 72 + 1 + 3 + 8 + 95 + 7 + 8983 + 8 + 3 + 36
= 72 + 1 + 3 + 8 + 95 + 7 + 8983 + 8 + 33 + 6 = 72 + 1 + 3 + 895 + 7898 + 3 + 8 + 336
= 72 + 1 + 3 + 8957 + 8 + 98 + 38 + 3 + 36 = 72 + 1 + 3 + 8957 + 8 + 98 + 38 + 33 + 6
= 72 + 1 + 38 + 95 + 7 + 8983 + 8 + 3 + 3 + 6 = 72 + 13 + 8 + 9 + 5 + 7 + 8983 + 83 + 36
= 72 + 13 + 89 + 5 + 7 + 8983 + 8 + 3 + 36 = 72 + 13 + 89 + 5 + 7 + 8983 + 8 + 33 + 6
= 72 + 13 + 8957 + 89 + 8 + 38 + 3 + 36 = 72 + 13 + 8957 + 89 + 8 + 38 + 33 + 6
= 721 + 38 + 9 + 5 + 7 + 8 + 9 + 83 + 8336 = 721 + 38 + 9 + 5 + 7 + 8 + 9 + 8383 + 36
= 721 + 38 + 9 + 5 + 7 + 89 + 8 + 3 + 8336 = 7213 + 8 + 95 + 78 + 983 + 833 + 6
= 7213 + 8 + 957 + 8 + 983 + 8 + 3 + 36 = 7213 + 8 + 957 + 8 + 983 + 8 + 33 + 6
= 7213 + 895 + 78 + 983 + 8 + 3 + 36 = 7213 + 895 + 78 + 983 + 8 + 33 + 6 = 962.
Page of Squares : First Upload March 22, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan