The first integer which is not the sum of two squares.
The first integer which is the sum of three squares.
The first integer which is the sum of a square and a prime : 12 + 2.
The smallest squares containing k 3's are :
36 = 62, 3136 = 562, 343396 = 5862, 1336336 = 11562,
133333209 = 115472, 3333330225 = 577352, 313341333361 = 5597692,
1433333333961 = 11972192, 43333334833636 = 65828062.
The sum of its divisors is 4, a square.
32 is the first square which is the sum of 3 squares: 12 + 22 + 22.
32 is the only square which is the sum of 2 consecutive cubes: 32 = 13 + 23.
33 = 27, 2 + 7 = 32; 35 = 243, 2 + 4 + 3 = 32.
Every integer of the form 8n + 3 is a sum of 3 non-zero squares.
An integer n can be the sum of at most 3 squares if and only if n is not of the form 4k(8a + 7).
The limit as x tends to infinity of N3(x)/x is 5/6, where N3(x) is the number of integers n < x which can be represented by sums of 3 squares.
(1/1)2 + (1/2)2 + (1/3)2 = (7/6)2
(12 + 22 + ... + 132) = 32 x (12 + 22 + ... + 62).
(1 + 2) x (3) = (1) x (2 + 3 + 4) = 32.
32 = (1 + 2)2 = 13 + 23.
1! + 2! + 3! = 32.
32± 2 are primes (the first case).
13 + 23 + 33 = (1 + 2 + 3)2,
In general, 13 + 23 + 33 + ... + n3 = (1 + 2 + 3 + ... + n)2
13 + 33 + 53 + ... + (2n - 1)3 = n2(2n2 - 1),
where nk+2 = 6nk+1 - nk, k = 0,1,2,..., n0 = 1, n1 = 5
Examples: 13 + 33 + 53 + 73 + 93 = 252, 13 + 33 + 53 + ... + 573 = 11892.
n3 - (n-1)3 + (n-2)3 - (n-3)3 + ... + 13 = a2, (n = (u2 + 1) / 2, u:odd),
Example: 53 - 43 + 33 - 23 + 13 = 92.
32 + 42 = 52,
32 + 42 + 52 + ... + 5802 = 80752,
32 + 42 + 52 + ... + 9632 = 172672.
(32 + 1) = (12 + 1 )(22 + 1 ),
(32 - 3) = (22 - 3 )(32 - 3 ).
33 = 27, 2 + 7 = 32,
34 = 81, 8 + 1 = 32,
35 = 243, 2 + 4 + 3 = 32,
330 = 205891132094649, 2052 + 892 + 112 + 32 + 22 + 0942 + 62 + 42 + 92 = 310.
Komachi Fractions: 32 = 57429/6381 = 58239/6471 = 75249/8361 = 95742/10638
= 95823/10647 = 97524/10836,
Komachi Fractions: (3/2)2 = 39285/17460 = 69417/30852, (3/4)2 = 50463/89712 = 52407/93168 = 53217/94608, (3/5)2 = 29403/81675, (3/16)2 = 3726/105984
Komachi equations:
32 = 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 305 equations,
32 = 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 311 equations,
32 = 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 202 equations,
32 = 12 - 22 * 32 * 42 / 562 * 72 - 82 + 92 = 12 + 22 + 32 + 42 - 52 + 62 - 72 - 82 + 92
= 12 + 22 + 32 * 42 * 52 / 62 + 72 - 82 - 92, and more 6 equations,
32 = 92 * 82 - 762 + 52 + 42 * 32 * 22 * 12 = 92 + 82 - 72 + 62 + 52 - 42 * 32 - 22 * 12
= 92 - 82 + 72 - 62 - 52 + 42 - 32 - 22 + 12, and more 9 equations,
32 = 982 / 72 + 62 + 52 - 42 * 32 - 22 - 102 = - 92 + 82 - 72 - 62 - 52 + 42 * 32 / 22 + 102
= - 92 + 82 - 72 - 62 * 52 / 42 / 32 * 22 + 102 = - 92 + 82 - 72 + 62 - 52 - 42 * 32 / 22 + 102,
32 = 13 * 23 - 33 / 43 * 563 / 73 - 83 + 93 = - 13 + 23 - 33 + 43 - 53 + 63 - 73 - 83 + 93,
32 = 93 - 83 - 73 + 63 - 53 + 43 - 33 + 23 - 13
Page of Squares : First Upload August 17, 2003 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan