The smallest squares containing k 30's :
2304 = 482, 300304 = 5482, 3030282304 = 550482,
3023060303025 = 17386952, 6300303630300304 = 793744522.
The first integer which is the sum of 6 squares in just 3 ways (see 28).
The first integer which is the sum of 15 squares in just two ways (see 20).
The first integer which is the sum of 4 distinct squares: 12 + 22 + 32 + 42.
The 5th integer which is the sum of 3 distinct squares: 12 + 22 + 52.
30 = 12 + 22 + 32 + 42.
302 = (1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9 + 10) = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11).
302 + 512 = 3501.
302 = 53 + 63 + 63 + 73.
302 = 3! + 3! + 4! + 4! + 5! + 6!
302 = (12 + 9)(92 + 9).
302 is the first square which is the sum of 6 4th powers in just 2 ways:
14 + 14 + 14 + 24 + 44 + 54 = 24 + 24 + 34 + 34 + 34 + 54.
The integral triangle of sides 17, 113, 120 has area 302, a square.
30k + 57k + 168k + 474k are squares for k = 1,2,3 (272, 5072, 105572).
21k + 45k + 201k + 633k are squares for k = 1,2,3 (302, 6662, 161822).
29k + 109k + 301k + 461k are squares for k = 1,2,3 (302, 5622, 112502).
Komachi equations:
302 = 1234 - 5 * 67 - 8 + 9 = 1 + 2 + 3 * 45 * 6 + 78 + 9
= 1 + 2 * 3 + 45 * 6 + 7 * 89, and more 2 equations,
302 = 9 + 876 + 5 + 4 + 3 + 2 + 1 = 9 + 876 + 5 + 4 + 3 * 2 * 1
= 9 + 876 + 5 + 4 * 3 - 2 * 1, and more 35 equations,
302 = 9 + 876 + 5 * 4 + 3 + 2 - 10 = 9 + 876 - 5 + 4 + 3 * 2 + 10
= 9 + 876 - 5 + 4 * 3 - 2 + 10, and more 16 equations,
302 = 12 * 22 + 32 * 42 * 52 - 62 * 782 / 92 = 12 + 232 + 42 * 52 + 62 - 72 + 82 - 92
= 122 * 32 - 42 * 52 + 62 - 72 - 82 + 92 = 122 / 32 + 42 + 52 * 62 - 72 - 82 + 92,
302 = 92 * 82 + 72 + 62 - 52 * 42 - 32 * 212 = 92 * 82 - 72 - 652 - 42 + 32 - 22 + 12
= 982 - 762 - 542 - 32 - 22 + 12,
302 = - 123 / 33 / 43 + 53 + 63 + 73 - 83 + 93.
The sum of the squares of aliquot divisors of 30 is 400, a square:
12 + 22 + 32 + 52 + 62 + 102 + 152 = 202
302 + 312 + 322 + ... + 1982 = 16122.
302 = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11) = (1)(2 + 3)(4)(5 + 6 + 7 + 8 + 9 + 10).
(1 + 2 + 3 + 4)(5)(6 + 7 + ... + 30) = 1502.
13 + 23 + ... + 303 = (1 + 2 + ... + 30)2 = 4652.
302 = 900 appears in the decimal expressions of π and e:
π = 3.14159•••900••• (from the 1188th digit),
(900 is the third 3-digit square in the expr. of e.)
e = 2.71828•••900••• (from the 139th digit).
Page of Squares : First Upload November 10, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan