The smallest squares containing k 14's :
144 = 122, 2214144 = 14882, 11414144569 = 1068372,
14014144141209 = 37435472, 1445143114141444 = 380150382.
The first integer which is the sum of 3 distinct squares: 12 + 22 + 32.
14 is the sum of m squares for m = 3,5,6,8.
14 = 12 + 22 + 32.
142 = 196, 16 = 42, 9 = 32, the first mosaic square.
An exchangeable square: 142 = 196, 961 = 312.
142 = 196, a zigzag square.
142 = (22 + 3)(52 + 3).
142 = 196, 1 + 9 + 6 = 42,
142 = 196, 19 + 6 = 52.
142± 3 are primes.
14k + 58k + 122k + 130k are squares for k = 1,2,3 (182, 1882, 20522).
14k + 1162k + 1274k + 2450k are squares for k = 1,2,3 (702, 29962, 1354362).
142 is the sum of m distinct squares for m = 3,4,5,6,7.
42 + 62 +122, 12 + 52 + 72 + 112, 12 + 32 + 42 + 72 + 112,
12 + 22 + 32 + 52 + 62 + 112 = 12 + 22 + 52 + 62 + 72 + 92 = 12 + 32 + 42 + 52 + 82 + 92,
12 + 22 + 32 + 42 + 62 + 72 + 92
142 is the first square which is the sum of 9 cubes in just 2 ways.
142 is the third square which is the sum of 6 fourth powers: 14 + 14 + 24 + 24 + 34 + 34.
(142 + 2) = (12 + 2)(82 + 2) = (32 + 2)(42 + 2) = (12 + 2)(22 + 2)(32 + 2),
(142 + 4) = (12 + 4)(62 + 4), (142 - 6) = (42 - 6)(52 - 6),
(142 + 8) = (22 + 8)(32 + 8).
Komachi Fractions: (14/17)2 = 15876/23409, (14/37)2 = 10584/73926,
(14/103)2 = 7056/381924
Komachi equations:
142 = 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 51 equations,
142 = 9 - 8 + 7 * 6 * 5 - 4 * 3 - 2 - 1 = 9 - 8 + 76 + 5 * 4 * 3 * 2 - 1
= 98 + 76 + 5 * 4 + 3 - 2 + 1, and more 50 equations,
142 = 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 54 equations,
142 = 122 - 32 - 42 - 52 + 62 + 72 - 82 + 92 = 12 / 22 * 342 - 52 - 62 - 72 - 82 + 92
= 12 - 22 + 32 * 42 + 52 - 62 + 72 - 82 + 92, and more 2 equations,
142 = - 92 - 872 / 62 + 52 + 432 / 22 * 12 = 982 / 72 + 62 - 52 - 42 + 32 - 22 * 12
= 92 - 82 + 72 - 62 + 52 + 42 * 32 - 22 + 12, and more 8 equations,
142 = 92 * 82 * 72 * 62 / 52 / 4322 * 102 = - 982 / 72 - 62 + 542 / 32 + 22 + 102
= 92 + 82 - 72 + 62 - 52 - 42 + 32 - 22 + 102, and more 3 equations,
142 = - 13 * 23 - 33 - 43 + 563 / 73 + 83 - 93.
(1 + 2 + ... + 6)(7 + 8 + ... + 14) = 422,
(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14) = 1082,
(1 + 2 + ... + 9)(10)(11 + 12 + ... + 14) = 1502,
(1 + 2 + ... + 4)(5 + 6 + ... + 10)(11 + 12 + ... + 14) = 1502,
(12 + 22 + ... + 72)(82 + 92 + ... + 142) = 3502,
(13 + 23 + ... + 143) = (1 + 2 + ... + 14)2 = 1052,
(13 + 23 + ... + 53)(63)(73 + 83 + ... + 143) = 226802,
(products of at most 3 factors)
148 = 1475789056, 1 + 4 + 7 + 5 + 78 + 90 + 5 + 6 = 1 + 4 + 75 + 7 + 8 + 90 + 5 + 6
= 1 + 47 + 5 + 78 + 9 + 056 = 14 + 75 + 7 + 89 + 05 + 6 = 142.
142 = 196 appears in the decimal expressions of π and e:
π = 3.14159•••196••• (from the 198th digit), 196 is the 3rd 3-digit square in the expr. of π,
e = 2.71828•••196••• (from the 1954th digit).
Page of Squares : First Upload September 15, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan