The smallest squares containing k 54's :
5476 = 742, 4545424 = 21322, 9545485401 = 977012,
545454625401 = 7385492, 35458542540054544 = 1883043882.
The first integer which is the sum of 3 squares in just 3 ways:
12 + 22 + 72 = 22 + 52 + 52 = 32 + 32 + 62.
The 6th integer which is the sum of 4 distinct squares: 22 + 32 + 42 + 52.
The 1st integer which is the sum of 6 squares in just 6 ways.
[1,1,1,1,1,7], [1,1,1,1,5,5], [1,1,2,4,4,4], [1,1,3,3,3,5], [1,2,2,2,4,5], [3,3,3,3,3,3]
The 2nd integer which is the sum of 7 squares in just 5 ways.
The 5th integer which is the sum of m squares in just 5 ways (m = 8, 9, 10).
54 = (12)(22 + 32 + 42 + 52).
542 = 2916 is a zigzag square with different digits.
542 = 3! + 3! + 4! + 6! + 6! + 6! + 6!.
542 = (12 + 5)(22 + 5)(72 + 5) = (22 + 2)(222 + 2) = (22 + 2)(42 + 2)(52 + 2).
66k + 210k + 1230k + 1410k are squares for k = 1,2,3 (542, 18842, 683642).
138k + 570k + 1086k + 1122k are squares for k = 1,2,3 (542, 16682, 536762).
210k + 606k + 762k + 1338k are squares for k = 1,2,3 (542, 16682, 554042).
Komachi Fraction : 542 = 813564/279.
Komachi equations:
542 = 98 * 7 * 6 - 5 * 4 * 3 * 2 * 10,
542 = 122 - 32 - 42 * 52 - 62 + 72 * 82 + 92 = 12 * 22 + 342 - 52 + 62 * 72 - 82 + 92
= 92 - 82 - 72 + 62 + 542 - 32 + 22 + 12 = - 92 - 82 + 72 * 62 + 542 / 32 * 22 + 12
= - 92 + 82 + 72 - 62 + 542 + 32 - 22 - 12 = - 92 + 82 * 72 + 62 * 52 - 42 - 322 + 12
= 92 + 82 * 72 - 62 - 52 - 42 * 32 + 22 - 102 = - 92 + 82 * 72 + 62 - 52 * 42 + 32 / 22 * 102.
(542 - 2) = (72 - 2)(82 - 2), (542 + 4) = (22 + 4)(192 + 4).
542 = (1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16),
542 = (1 + 2)(3)(4)(5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13).
542 is the third squares which is the sum of 4 sixth powers : 36 + 36 + 36 + 36,
542 is the third squares which is the sum of 8 fourth powers :
14 + 24 + 24 + 34 + 44 + 54 + 54 + 64.
542 = 12 + 152 + 292 + 432.
542 = 2916, 29 + 1 + 6 = 62.
548 = 72301961339136, 7 + 23 + 01961 + 3 + 3 + 913 + 6 = 542,
549 = 3904305912313344, 39 + 0430 + 5 + 91 + 2313 + 34 + 4 = 542.
(12 + 22 + 32 + ... + 542) = 53955, which consists of (3 kinds of) odd digits (the fourth 5-digit.)
(1)(2 + 3 + ... + 26)(27 + 28 + ... + 54) = 6302,
(1 + 2 + ... + 12)(13 + 14 + ... + 26)(27 + 28 + ... + 54) = 49142,
(1 + 2 + ... + 15)(16 + 17 + ... + 29)(30 + 31 + ... + 54) = 63002.
542 = 2916 appears in the decimal expressions of π and e:
π = 3.14159•••2916••• (from the 3368th digit),
e = 2.71828•••2916••• (from the 9678th digit).
Page of Squares : First Upload January 13, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan