The smallest squares containing k 42's :
4225 = 652, 2424249 = 15572, 4242177424 = 651322,
14224242422016 = 37715042, 4242446423424225 = 651340652.
42 is the sum of m squares for m = 3, 4, ..., 28.
The 9th integer which is the sum of 3 distinct squares: 12 + 42 + 52.
The 2nd integer which is the sum of 4 squares in just 3 ways: [1,1,2,6], [1,3,4,4], [2,2,3,5].
The 3rd integer which is the sum of 7 squares in just 4 ways (see 41).
The 1st integer which is the sum of 12 squares in just 5 ways.
422 = 1764, a square with different digits.
422 = 3*4*5 + 5*6*7 + 7*8*9 + 9*10*11.
422 = (12 + 5)(172 + 5) = (12 + 5)(32 + 5)(42 + 5) = (32 + 3)(122 + 3) = (32 + 5)(112 + 5).
422 = 1764, 1 + 76 + 4 = 92, 422 = 1764, 17 + 64 = 92.
72 + 152 + 232 + 312 = 422.
422 = 13 + 23 + 33 + 123 = 13 + 63 + 63 + 113 = 23 + 33 + 93 + 103.
The sum of the squares of the divisors of 42 is a square, 502.
Loop of length 8 by the function f(N) = ... + c2 + b2 + a2 for N = ... + 102c + 10b + a:
42 -- 20 -- 4 -- 16 -- 37 -- 58 -- 145 -- 42
42k + 66k + 108k + 145k are squares for k = 1,2,3 (192, 1972,21612).
42k + 129k + 660k + 1194k are squares for k = 1,2,3 (452, 13712, 446312).
178k + 242k + 494k + 850k are squares for k = 1,2,3 (422, 10282, 274682).
Komachi Fractions : (19/42)2 = 3249/15876=6498/31752=16245/79380.
Komachi equations:
422 = 1 + 23 * 4 + 5 * 6 * 7 * 8 - 9 = 1 + 2 - 3 + 4 * 56 * 7 / 8 * 9
= 123 * 4 * 5 + 6 - 78 * 9 = - 1 - 2 + 3 + 4 * 56 * 7 / 8 * 9,
422 = 987 + 65 * 4 * 3 - 2 - 1 = 9 * 87 + 654 * 3 / 2 * 1
= 9 * 8 * 7 * 6 - 5 * 4 * 3 * 21, and more 2 equations,
422 = - 9 * 8 + 7 * 65 * 4 + 3 * 2 + 10 = - 9 * 8 + 765 * 4 * 3 * 2 / 10
= - 98 + 7 * 65 * 4 + 32 + 10,
422 = 122 * 32 + 42 * 52 + 62 + 72 + 82 - 92,
422 = - 93 - 83 - 73 - 63 + 53 + 43 + 33 / 23 * 103.
422 = 3 x 4 x 5 + 5 x 6 x 7 + 7 x 8 x 9 + 9 x 10 x 11.
422 = (1)(2)(3)(4 + 5 + ... + 24) = (1)(2)(3 + 4)(5 + 6 + ... + 16) = (1)(2)(3 + 4 + ... + 9)(10 + 11)
= (1 + 2 + ... + 6)(7 + 8 + ... + 14) = (1 + 2 + ... + 7)(8 + 9 + ... + 13) = (1 + 2 + 3)(4 + 5 + ... + 24),
422 = (12 + 22 + 32)(42 + 52 + 62 + 72).
(1 + 2 + ... + 25)(26 + 27 + ... + 38)(39 + 40 + ... + 42) = 46802,
(1 + 2 + ... + 36)(37)(38 + 39 + ... + 42) = 22202,
(12 + 22 + ... + 72)(82 + 92 + 102)(112 + 122 + ... + 422) = 294002,
(12 + 22 + ... + 82)(92 + 102 + ... + 322)(332)(342 + 352 + ... + 422) = 57087362,
(12 + 22 + ... + 132)(142 + 152 + ... + 382)(392 + 402 + 412)(422) = 112366802.
428 = 9682651996416, 9 + 6 + 82 + 651 + 996 + 4 + 16 = 422,
9 + 682 + 6 + 51 + 996 + 4 + 16 = 9 + 682 + 65 + 1 + 996 + 4 + 1 + 6
= 96 + 8 + 2 + 651 + 996 + 4 + 1 + 6 = 968 + 265 + 19 + 96 + 416 = 422,
429 = 406671383849472,
4 + 066 + 713 + 83 + 849 + 47 + 2 = 40 + 6 + 6 + 7 + 1 + 383 + 849 + 472
= 40 + 6 + 6 + 713 + 8 + 38 + 4 + 947 + 2 = 40 + 6 + 671 + 3 + 8 + 3 + 84 + 947 + 2
= 40 + 6 + 671 + 3 + 83 + 8 + 4 + 947 + 2 = 40 + 66 + 713 + 83 + 849 + 4 + 7 + 2
= 40 + 66 + 713 + 838 + 4 + 94 + 7 + 2 = 40 + 667 + 13 + 8 + 3 + 84 + 947 + 2
= 40 + 667 + 13 + 83 + 8 + 4 + 947 + 2 = 406 + 6 + 7 + 1 + 3 + 8 + 384 + 947 + 2
= 406 + 6 + 7 + 1 + 3 + 838 + 494 + 7 + 2 = 406 + 6 + 7 + 1 + 383 + 8 + 4 + 947 + 2
= 406 + 6 + 7 + 13 + 8 + 3 + 849 + 472 = 406 + 6 + 71 + 383 + 849 + 47 + 2 = 422.
Page of Squares : First Upload December 22, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan