The smallest squares containing k 31's :
3136 = 562, 1313316 = 11462, 13183173124 = 1148182,
9313123131081 = 30517412, 13131313171331364 = 1145919422.
The first integer which is the sum of 4 squares in just 2 ways:
12 + 12 + 22 + 52 = 22 + 32 + 32 + 32
The first integer which is the sum of 7 squares in just 3 ways (see 28).
312 is the second square which is sum of 7 fourth powers : 24 + 24 + 24 + 24 + 24 + 44 + 54.
312 = 23 + 23 + 63 + 93.
312 = 961 with a decreasing sequence of digits.
312 = 961, a reversible square (169 = 132).
312 = 961, an exchangeable square (196 = 142).
312 = 961, 9 + 6 + 1 = 42.
312 = 1! + 5! + 5! + 6!
The sum of odd consecutive primes 3 + 5 + 7 + 11 + 13 + 17 + ... + 89 = 312.
72 + 152 + 232 + 312 = 422.
31 and 32 is the first pair of consecutive happy numbers.
A happy number ... Start with an integer n. Square and add its digits. Repeat these steps. If you eventually obtain 1, then n is a happy number.
315 = 28629151, 2 + 862 + 91 + 5 + 1 = 312,
318 = 852891037441, 8 + 5 + 2 + 891 + 03 + 7 + 4 + 41 = 312,
8 + 5 + 2 + 891 + 03 + 7 + 44 + 1 = 8 + 528 + 9 + 1 + 0374 + 41
= 852 + 8 + 9 + 10 + 3 + 74 + 4 + 1 = 852 + 8 + 9 + 10 + 37 + 4 + 41
= 852 + 8 + 9 + 10 + 37 + 44 + 1 = 852 + 89 + 1 + 03 + 7 + 4 + 4 + 1 = 312.
60k + 241k + 282k + 378k are squares for k = 1,2,3 (312, 5332, 95212).
98k + 212k + 305k + 346k are squares for k = 1,2,3 (312, 5172, 89592).
Komachi Fractions : (6/31)2 = 4860/129735, (8/31)2 = 8640/129735,
(22/31)2 = 13068/25947, (31/45)2 = 17298/36450, (31/63)2 = 8649/35721,
(31/285)2 = 8649/731025.
Komachi equations:
312 = 12 / 3 * 4 * 56 + 7 * 8 + 9 = 12 / 3 * 4 * 56 - 7 + 8 * 9
= 1 + 23 * 45 - 6 - 78 + 9, and more 4 equations,
312 = 987 - 6 + 5 - 4 * 3 * 2 - 1 = 987 - 6 - 5 - 4 * 3 - 2 - 1
= 987 - 6 - 5 * 4 + 3 - 2 - 1, and more 15 equations,
312 = 987 + 6 - 5 - 4 - 3 - 2 * 10 = 987 + 6 * 5 + 4 - 3 * 2 * 10
= 987 - 6 - 5 - 4 - 3 + 2 - 10, and more 17 equations,
312 = 12 * 22 + 32 + 42 + 52 * 62 + 72 + 82 - 92 = 12 + 22 / 32 * 452 - 62 - 72 + 82 + 92,
312 = 92 * 82 - 72 - 652 + 42 + 32 * 22 - 12 = 92 / 82 * 72 + 62 * 52 - 42 / 322 - 12
= 92 + 82 - 72 + 62 * 52 - 42 * 32 / 22 + 12 = 92 + 82 - 72 - 62 + 52 * 42 * 32 / 22 + 12,
312 = 13 + 23 + 33 - 43 - 53 + 63 - 73 + 83 + 93 = 123 + 33 - 43 - 563 / 73 / 83 - 93,
312 = 93 + 83 - 73 + 63 - 53 - 43 + 33 + 23 + 13,
312 = 14 + 24 + 34 + 44 - 54 + 64 + 74 + 84 - 94 = - 94 + 84 + 74 + 64 - 54 + 44 + 34 + 24 + 14.
(312 - 1) = (32 - 1)(112 - 1), (312 + 1) = (52 + 1)(62 + 1),
(312 + 5) = (32 + 5)(82 + 5), (312 + 7) = (22 + 7)(92 + 7),
(312 - 7) = (32 - 7)(222 - 7), (312 + 8) = (32 + 8)(72 + 8).
(1 + 2 + 3)(4)( 5 + ... + 31) = 1082,
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 31) = 5882,
(1 + 2 + ... + 12)(13 + 14 + ... + 20)(21 + 22 + ... + 31) = 17162.
13 + 23 + ... + 313 = (1 + 2 + ... + 31)2 = 4962.
312 = 961 appears in the decimal expressions of π and e:
π = 3.14159•••961••• (from the 1235th digit),
e = 2.71828•••961••• (from the 701st digit) (961 is the 10th 3-digit square in the expr. of e.)
Page of Squares : First Upload November 10, 2003 ; Last Revised January 27, 2011
by Yoshio Mimura, Kobe, Japan