The smallest squares containing k 27's :
2704 = 522, 2702736 = 16442, 9272727025 = 962952,
2727927902736 = 16516442, 12727427272704 = 35675522.
The first integer which is the sum of a square and a prime in just 3 ways :
22 + 23, 42 + 11, 52 + 2.
The first integer which is the sum of 3 squares in just two ways:
27 = 12 + 12 + 52 = 32 + 32 + 32.
The smallest integer that is the sum of 3 squares in two ways: 27 = 32 + 32 + 32 = 12 + 12 + 52.
272 = 729, 7 + 29 = 62,
272 = 729, 72 + 9 = 92.
272 = 729 is a zigzag square.
272 = 1! + 2! + 3! + 6!
272 = 13 + 63 + 83 = 35 + 35 + 35.
272 = ( 2 + 7 )3.
27 and 28 are consecutive integers having square factors (the third case).
22k + 161k + 266k + 280k are squares for k = 1,2,3 (272, 4192, 67052).
30k + 57k + 168k + 474k are squares for k = 1,2,3 (272, 5072, 105572).
52k + 190k + 233k + 254k are squares for k = 1,2,3 (272, 3972, 60032).
57k + 102k + 222k + 348k are squares for k = 1,2,3 (272, 4292, 73712).
Komachi Fraction : (4/27)2 = 7056/321489.
Komachi equations:
272 = 1 + 2 * 3 * 4 * 5 * 6 + 7 - 8 + 9 = 1234 + 5 - 6 - 7 * 8 * 9
= 1 * 234 + 567 - 8 * 9, and more 24 equations,
272 = 9 + 8 - 7 + 6 * 5 * 4 * 3 * 2 - 1 = 9 - 8 + 7 + 6 * 5 * 4 * 3 * 2 + 1
= 98 * 7 + 6 * 5 + 4 * 3 + 2 - 1, and more 31 equations,
272 = 98 * 7 + 6 + 5 + 4 * 3 + 2 * 10 = 98 * 7 + 6 + 5 * 4 - 3 + 2 * 10
= 98 * 7 + 6 * 5 + 4 - 3 + 2 + 10, and more 30 equations,
272 = 123 - 33 * 43 + 563 - 73 * 83 + 93 = 123 - 33 * 43 + 563 / 73 - 83 + 93
= - 123 + 33 * 43 - 563 + 73 * 83 + 93, and more 33 equations,
272 = 12 + 22 - 32 * 42 + 52 * 62 - 72 - 82 + 92 = 12 + 232 + 42 + 52 - 62 + 72 + 82 + 92,
272 = 92 + 872 - 62 - 542 - 32 * 212 = 92 - 82 - 72 + 62 * 52 - 42 * 32 + 22 + 12.
277 = 10460353203, 10 + 460 + 3 + 53 + 203 = 272,
278 = 282429536481, 2 + 8 + 2 + 4 + 2 + 9 + 53 + 648 + 1
= 28 + 242 + 9 + 5 + 364 + 81 = 282 + 4 + 295 + 3 + 64 + 81 = 272,
279 = 7625597484987, 7 + 6 + 2 + 5 + 597 + 4 + 8 + 4 + 9 + 87 = 272 and more 15 equations,
2710 = 205891132094649, 2 + 05 + 8 + 9 + 11 + 32 + 09 + 4 + 649 = 272 and more 18 equations,
2711 = 5559060566555523, 5 + 5 + 5 + 9 + 06 + 05 + 6 + 655 + 5 + 5 + 23 = 272 and more 82 equations.
(272 - 3) = (52 - 3)(62 - 3), (272 + 7) = (42 + 7)(52 + 7).
272 + 282 + 292 + ... + 592 = 2532,
272 + 282 + 292 + ... + 3642 = 40172.
12 + 22 + 32 + ... + 272 = 6930, which consists of different digits.
272 = (1)( 2 + 3 + 4 + 5 + 6 + 7)(8 + 9 + 10),
272 = (1)( 2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13).
(1 + 2 + 3 )(4 + 5 + ... + 12)(13 + 14 + ... + 27) = 3602,
(12 + 22 + 32)(42 + 52 + ... + 72)(82 + 92 + ... + 132)(142 + 152 + ... + 272) = 855542,
(12 + 22 + 32)(42 + 52 + ... + 72)(82 + 92 + ... + 142)(152 + 162 + ... + 272) = 955502,
(12 + 22 + 32)(42 + 52 + ... + 132)(142 + 152 + ... + 172)(182 + 192 + ... + 272) = 2366702.
13 + 23 + ... + 273 = (1 + 2 + ... + 27)2 = 3782.
272 = 729 appears in the decimal expressions of π and e:
π = 3.14159•••729••• (from the 770th digit), (729 is the 7th 3-digit square in the expr. of π)
e = 2.71828•••729••• (from the 137h digit), (729 is the 2nd 3-digit square in the expr. of e).
Page of Squares : First Upload October 27, 2003 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan