The second number which is the sum of 4 squares: 12 + 12 + 12 + 22.
The first integer which is not the sum of less than four squares. The next are 15, 23, 28, 31, ...
The smallest squares containing k 7's are :
576 = 242, 5776 = 762, 179776 = 4242, 12787776 = 35762,
77774761 = 88192, 7907477776 = 889242, 77770707876 = 2788742,
4477777637776 = 21160762, 77777477275716 = 88191542.
Every integer of the form 4k(8m + 7) is not the sum of three squares.
The third integer which is the sum of a square and a prime in just one way : 22 + 3.
72 = 49, the fourth square with an increasing sequence of digits.
The Fermat quotient (2p-1-1)/p is a square only when p = 3 or p = 7.
25 - 7 = 52, 27 - 7 = 112, 215 - 7 = 1812.
70 + 71 + 72 + 73 = 202.
7 = (12 + 22 + 32 + ... + 102) / (12 + 22 + 32 + 42 + 52).
7! + 1 = 712, 72 = 1! + 4! + 4!.
72 is the first integer which is the sum of 3 distinct squares: 22 + 32 + 62.
72 is the first integer which is the sum of 5 squares in just 2 ways:
12 + 22 + 22 + 22 + 62 = 22 + 22 + 32 + 42 + 42.
72 is the second integer which is the sum of 4 squares in just 2 ways:
12 + 42 + 42 + 42 = 22 + 22 + 42 + 52.
72 is the third integer which is the sum of 3 squares: 22 + 32 + 62.
72 is the second integer which is the sum of 4 fourth powers: 14 + 24 + 24 + 24.
72 = 49, where 4 and 9 are squaes.
72 + 82 + 92 + 102 + ... + 292 = 922,
72 + 82 + 92 + 102 + ... + 392 = 1432,
72 + 82 + 92 + 102 + ... + 562 = 2452,
72 + 82 + 92 + 102 + ... + 1902 = 15182,
72 + 82 + 92 + 102 + ... + 22152 = 602072.
(12 + 22 + 32) x (42 + 52 + 62 + 72) = 422,
13 + 23 + 33 + 43 + 53 + 63 + 73 = (1 + 2 + 3 + 4 + 5 + 6 + 7)2,
(13) x (23 + 33) x (43 + 53 + 63) x (73) = 22052.
(72 + 1) = (22 + 1)(32 + 1),
(72 + 5) = (12 + 5)(22 + 5), (72 - 5) = (32 - 5)(42 - 5).
Cubic polynomial (X + 72)(X + 1082)(X + 3362) = X3 + 3532X2 + 363722X + 2540162.
711 = 1977326743, 1 + 9 + 7 + 7 + 3 + 2 + 6 + 7 + 4 + 3 = 72.
Komachi Fractions: 72 = 197568/4032, (7/11)2 = 15876/39204.
Komachi equations:
72 = 1 + 2 + 3 * 4 * 5 - 6 - 7 + 8 - 9 = 1 + 2 - 3 * 4 + 5 + 6 + 7 * 8 - 9
= 1 + 2 * 3 + 4 + 5 * 6 + 7 - 8 + 9, and more 263 equations,
72 = 9 + 8 + 7 + 6 + 5 + 4 * 3 + 2 * 1 = 9 + 8 + 7 + 6 + 5 * 4 - 3 + 2 * 1
= 9 + 8 + 7 + 6 - 5 + 4 * 3 * 2 * 1, and more 289 equations,
72 = 9 + 8 + 7 + 6 + 5 + 4 * 3 * 2 - 10 = 9 + 8 + 7 + 6 - 5 + 4 * 3 + 2 + 10
= 9 + 8 + 7 + 6 * 5 + 4 + 3 - 2 - 10, and more 174 equations,
72 = 12 / 22 + 32 - 452 / 62 - 72 + 82 + 92 = 12 * 22 * 32 - 42 + 52 + 62 - 72 - 82 + 92
= - 122 / 32 - 42 - 562 + 72 * 82 + 92, and more 12 equations,
72 = 92 - 82 - 72 - 62 - 542 / 32 + 212 = 92 + 82 - 72 - 62 + 52 - 42 * 32 / 22 * 12
= 92 - 82 + 72 + 62 - 52 - 42 - 32 - 22 + 12, and more 5 equations,
72 = 92 + 82 + 72 + 62 - 542 / 32 / 22 - 102 = 982 / 72 - 62 + 52 - 42 * 32 / 22 - 102
= 92 * 82 + 72 - 62 * 52 * 42 * 32 * 22 / 102, and more 17 equations,
72 = 13 - 23 + 33 + 43 - 53 + 63 - 73 - 83 + 93,
72 = 93 - 83 - 73 + 63 - 53 + 43 + 33 - 23 + 13.
72 = 49 appears in the decimal expressions of π and e:
π = 3.14159•••49••• (from the 57th digit), 49 is the 3rd 2-digit square in the expr. of π,
e = 2.71828•••49••• (from the 34th digit), 49 is the 3rd 2-digit square in the expr. of e.
Page of Squares : First Upload August 24, 2003 ; Last Revised May 18, 2010
by Yoshio Mimura, Kobe, Japan