The smallest squares containing k 13's :
1369 = 372, 1313316 = 11462, 1131313225 = 336352,
1318131313801 = 11480992, 13113413138771344 = 1145138122.
The 3rd integer which is the sum of 2 distinct squares: 22 + 32.
13 = (12) x (22 + 32).
132 is the third square which is the sum of 2 distinct squares: 52 + 122.
132 is the 4th square which is the sum of 3 distinct squares.
132 is the first square which is the sum of 4 distinct squares in just 2 ways:
12 + 22 + 82 + 102 = 22 + 42 + 72 + 102.
132 is the 4th square which is the sum of 5 distinct squares: 22 + 42 + 62 + 72 + 82.
132 is the first square which is the sum of 6 distinct squares: 12 + 22 + 32 + 52 + 72 + 92.
132 is the 5th square which is the sum of 3 squares: 32 + 42 + 122.
132 is the second square which is the sum of 5 cubes: 13 + 23 + 23 + 33 + 53.
9k + 10k + 60k + 90k are squares for k = 1,2,3 (132, 1092, 9732).
10k + 44k + 57k + 58k are squares for k = 1,2,3 (132, 932, 6832).
22k + 24k + 58k + 65k are squares for k = 1,2,3 (132, 932, 7032).
132 = 1! + 4! + 4! + 5!
132 = 169, 1 + 6 + 9 = 42,
132 = 169, 16 + 9 = 52.
132 = 169 is a reverible square (961 = 312).
132 = 169 where 16 = 42 and 9 = 32.
132 = 169 with an increasing sequence of digits.
Komachi Fractions: 132 = 891306/5274, (11/13)2 = 67518/94302,
(12/13)2 = 42768/50193, (13/29)2 = 10647/52984, (13/95)2 = 9126/487350,
(13/125)2 = 9126/843750, (13/126)2 = 9126/857304
Komachi equations:
132 = 1 - 2 + 3 + 4 * 5 * 6 + 7 * 8 - 9 = 12 * 3 * 4 - 5 + 6 + 7 + 8 + 9
= 12 * 3 * 4 - 5 - 6 * 7 + 8 * 9, and more 62 equations,
132 = 98 + 7 + 6 + 5 * 4 * 3 - 2 * 1 = 98 + 7 * 6 + 5 + 4 * 3 * 2 * 1
= 98 + 7 * 6 + 5 * 4 * 3 / 2 - 1, and more 64 equations,
132 = 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 63 equations,
132 = 12 + 22 - 32 + 42 + 52 + 62 - 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,
132 = 92 + 82 / 72 * 62 + 52 + 42 - 32 / 212 = - 982 / 72 + 62 + 542 / 32 + 22 + 12
= 92 + 82 + 72 + 62 - 52 - 42 * 32 / 22 * 12, and more 14 equations,
132 = 982 / 72 - 62 + 52 * 42 + 32 - 22 * 102 = 982 / 72 - 62 + 52 * 42 * 32 / 22 / 102
= 982 / 72 - 62 - 52 * 42 + 32 + 22 * 102 = 92 * 82 * 72 * 652 / 42 / 32 / 2102,
132 = - 13 - 23 + 33 - 43 + 53 + 63 - 73 - 83 + 93,
132 = 93 - 83 - 73 + 63 + 53 - 43 + 33 - 23 - 13.
(132 + 1) = (32 + 1)(42 + 1) = (12 + 1)(22 + 1)(42 + 1),
(132 + 7) = (22 + 7)(32 + 7), (132 - 7) = (42 - 7)(52 - 7).
132 + 142 + 152 + ... + 1082 = 6522.
133 - 123 + 113 - 103 + 93 - 83 + 73 - 63 + 53 - 43 + 33 - 23 + 13 = 352.
132 = 83 - 73, the smallest square that is the difference of two cubics.
15 + 25 + 35 + 45 + 55 + 65 + 75 + 85 + 95 + 105 + 115 + 125 + 135 = 10012.
(1)(2 + 3 + 4)(5 + 6 + ... + 13) = 272,
(1 + 2 + ... + 7)(8 + 9 + ... + 13) = 422,
(13 + 23 + ... + 133) = (1 + 2 + ... + 13)2 = 912,
(products of at most 3 factors)
135 = 371293, 3 + 71 + 2 + 93 = 37 + 129 + 3 = 132,
138 = 815730721, 8 + 1 + 57 + 30 + 72 + 1 = 8 + 15 + 73 + 072 + 1
= 81 + 5 + 7 + 3 + 072 + 1 = 81 + 5 + 73 + 07 + 2 + 1 = 81 + 57 + 3 + 07 + 21 = 132.
1311 = 1792160394037, 1 + 7 + 9 + 2 + 1 + 60 + 3 + 9 + 40 + 37 = 132 and more 14 equations.
1314 = 3937376385699289, 3 + 9 + 3 + 7 + 3 + 7 + 6 + 3 + 8 + 5 + 6 + 9 + 9 + 2 + 89 = 132 and more 13 equations.
132 = 169 appears in the decimal expressions of π and e:
π = 3.14159•••169••• (from the 40th digit), 169 is the 2nd 3-digit square in the expr. of π,
e = 2.71828•••169••• (from the 399th digit), 169 is the 5th 3-digit square in the expr. of e.
Page of Squares : First Upload September 8, 2003 ; Last Revised January 25, 2011
by Yoshio Mimura, Kobe, Japan