The second number which is the sum of 3 squares: 12 + 12 + 22.
The first integer which is the sum of a square and a prime in just 2 ways : 12 + 5, 22 + 2.
The smallest squares containing k 6's are :
16 = 42, 676 = 262, 46656 = 2162, 1666681 = 12912,
26666896 = 51642, 4666665969 = 683132, 26661664656 = 1632842,
616686666436 = 7852942, 12666666068676 = 35590262.
The squares which begin with 6 and end in 6 are
676 = 262, 60516 = 2462, 64516 = 2542, 65536 = 2562, 69696 = 2642,...
For each k = 6, 7, 8, ..., every integer n is the sum of k non-zero squares unless n = 1, 2, ..., k-1, k+1, k+2, k+4, k+5, k+7, k+10, k+13.
62 = 36, 3 + 6 = 32.
62 = 3! + 3! + 4!.
62± 5 are primes.
62 is the first square which is the sum of 4 squares in just 2 ways:
12 + 12 + 32 + 52 = 32 + 32 + 32 + 32.
62 is the first square which is the sum of m squares in just 4 ways for m = 6, 9.
62 is the second square which is the sum of 3 squares: 22 + 42 + 42.
62 is the first square which is the sum of 3 distinct cubes: 13 + 23 + 33.
Using the first 9 digits of PI = 3.141592653...,
33 + 13 + 43 + 13 + 53 + 93 + 23 + 63 + 53 = (3 + 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5)2 = 64.
62 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = (1 + 2 + 3)2 = 13 + 23 + 33,
62 = (1 + 2) x (3) x (4) = (1 + 2) x (3 + 4 + 5) = (1) x (2) x (3 + ... + 6).
(1) x (2) x (3 + 4 + 5 + 6) = 62,
(1) x (2) x (3 + 4 + 5) x (6) = 122,
(1 + 2 + 3) x (4 + 5) x (6) = (1) x (2) x (3) x (4 + 5) x (6) = 182,
13 + 23 + 33 + 43 + 53 + 63 = (1 + 2 + 3 + 4 + 5 + 6)2,
(13 + 23) x (33 + 43 + 53) x (63) = 6482.
(15) x (25) x (35 + 45 + 55 + 65) = 6242.
6k + 408k + 690k + 921k are squares for k = 1,2,3 (452, 12212, 343172).
(62 + 4) = (12 + 4)(22 + 4), (62 - 6) = (32 - 6)(42 - 6).
62 = 36, the third square with an increasing sequence of digits.
64 = 1296, 1 + 29 + 6 = 62,
67 = 279936, 2 + 7 + 9 + 9 + 3 + 6 = 62,
68 = 1679616, 1 + 6 + 7 + 9 + 6 + 1 + 6 = 62,
69 = 10077696, 1 + 007 + 7 + 6 + 9 + 6 = 62,
610 = 60466176, 6 + 04 + 6 + 6 + 1 + 7 + 6 = 62,
613 = 13060694016, 1 + 3 + 06 + 06 + 9 + 4 + 01 + 6 = 62,
613 = 13060694016, 12 + 32 + 062 + 062 + 92 + 42 + 012 + 62 = 63.
Both 6 and 62 are triangular numbers.
6 and 7 are the only consecutive integers whose sums of the squares of their divisors are equal.
(12 + 22 + 33) + (42 + 52 + 63) = 172
(14 + 22 + 33) + (44 + 52 + 63) = 232
(19 + 27 + 37) + (49 + 57 + 67) = 7892
6! + 32 = 272
6! = 122 + 242 = 3! + 3! + 4!
62 = 82 - 72 + 62 - 52 + 42 - 32 + 22 - 12
12 + 22 + 32 + 42 + 52 + 62 = 91, which consists of odd digits.
1-2 + 2-2 + 3-2 + 4-2 + ... = Pi2 / 6
The integral triangles with the smallest square areas :
the area is 62, the sides are 3,25,26 (or 9,10,17).
Komachi Fractions: (5/6)2 = 27864/19350, (6/31)2 = 4860/129735,
(6/53)2 = 3564/278091 = 4860/379215, (6/233)2 = 972/1465803,
(6/851)2 = 324/6517809, (6/977)2 = 324/8590761
Komachi equations:
62 = 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 276 equations,
62 = 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 296 equations,
62 = 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 182 equations,
62 = 12 - 22 * 342 + 52 + 672 + 82 + 92 = 122 + 32 + 42 + 52 + 62 - 72 - 82 - 92
= 12 - 22 - 32 * 42 + 52 - 62 + 72 + 82 + 92, and more 8 equations,
62 = 982 / 72 - 62 + 52 - 42 * 32 - 22 - 12 = 92 - 82 - 72 + 62 - 52 * 42 - 32 + 212
= 92 + 82 + 72 - 62 + 52 - 42 * 32 - 22 + 12, and more 3 equations,
62 = 92 * 82 * 72 * 62 * 52 / 42 / 32 / 2102 = 92 + 82 + 72 - 62 - 52 + 42 - 32 - 22 - 102
= 92 - 82 - 72 - 62 + 52 - 42 - 32 + 22 + 102, and more 11 equations,
62 = 123 / 33 / 43 + 53 - 63 + 73 + 83 - 93.
62 = 36 appears in the decimal expressions of π and e:
π = 3.14159•••36••• (from the 285th digit), 25 is the 6th 2-digit square in the expr. of π,
e = 2.71828•••36••• (from the 19th digit), 25 is the 2nd 2-digit square in the expr. of e.
Page of Squares : First Upload August 24, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan