34 is the sum of m squares for m = 2, 3, ... , 20
The smallest squares containing k 34's :
3481 = 592, 5345344 = 23122, 34034346256 = 1844842,
3446343483489 = 18564332, 2743493434343424 = 523783682.
The 2nd integer which is the sum of 4 squares in just 2 ways: [1, 1, 4, 4], [1, 2, 2, 5]
The 2nd integer which is the sum of 7 squares in just 3 ways (see 32).
The 1st integer which is the sum of 10 squares in just 4 ways (see 32).
342 is the first square which is the sum of 7 fourth powers in just 2 ways:
14 + 14 + 14 + 24 + 44 + 44 + 54 = 24 + 24 + 34 + 34 + 34 + 44 + 54.
342 = 1156 is the first square of 4 digits which form an non-decreasing sequence : 1,1,5,6.
341 + 21341 + 28731 = 712, 342 + 21342 + 28732 = 35792, 343 + 21343 + 28733 = 1828452 (See 71).
342 = 1156, 115 + 6 = 112.
342 = (12 + 22 + 32 + ... + 112) + (12 + 22 + 32 + ... +122).
34k + 82k + 98k + 110k are squares for k = 1,2,3 (182, 1722, 16922).
33k + 205k + 441k + 477k are squares for k = 1,2,3 (342, 6822, 142462).
125k + 241k + 305k + 485k are squares for k = 1,2,3 (342, 6342, 125862).
186k + 234k + 258k + 478k are squares for k = 1,2,3 (342, 6202, 120682).
Komachi Fraction : (3/34)2 = 4617/593028.
Komachi equations:
342 = 1 * 234 * 5 - 6 - 7 + 8 - 9 = 1234 - 5 + 6 - 7 - 8 * 9
342 = 1 * 23 * 4 * 5 - 6 + 78 * 9, and more 2 equations,
342 = 987 + 6 + 54 * 3 + 2 - 1 = 9 * 8 * 7 + 654 - 3 + 2 - 1
= 9 + 8 * 7 + 6 + 543 * 2 - 1, and more 2 equations,
342 = 12 + 22 + 342 + 52 + 62 - 72 + 82 - 92 = 122 + 32 * 42 + 52 * 62 - 72 - 82 + 92,
342 = 92 * 82 + 72 - 62 - 52 * 42 * 32 - 212 = 92 * 82 + 72 - 652 + 42 * 32 + 22 * 12
= 92 * 82 + 72 - 652 + 42 * 32 + 22 / 12 = 92 - 82 + 72 * 62 + 52 * 42 - 322 - 12
= 92 - 82 - 72 * 62 + 542 - 32 - 22 * 12 = 92 - 82 - 72 * 62 + 542 - 32 - 22 / 12,
342 = 13 * 23 - 33 - 43 + 53 + 63 - 73 + 83 + 93 = 93 + 83 - 73 + 63 + 53 - 43 - 33 + 23 * 13
= 93 + 83 - 73 + 63 + 53 - 43 - 33 + 23 / 13.
(342 + 4) = (52 + 4)(62 + 4) = (12 + 4)(22 + 4)(52 + 4),
(342 - 4) = (42 - 4)(102 - 4), (342 - 6) = (42 - 6)(112 - 6),
(342 - 8) = (62 - 8)(72 - 8).
(1 + 2 + ... + 7)(8 + 9 + ... + 34) = 1262,
(1 + 2 + 3)(4 + 5 + ... + 15)(16 + 17 + ... + 34) = 5702.
13 + 23 + ... + 343 = (1 + 2 + ... + 34)2 = 5952,
(13 + 23 + ... + 133)(143 + 153 + ... + 343) = 535082.
12 + 22 + 32 + 42 + ... + 342 = 13685, which consists of different digits (the first 5-digit).
348 = 1785793904896, 1 + 7 + 8 + 57 + 93 + 90 + 4 + 896 = 342,
1 + 7 + 8 + 57 + 939 + 048 + 96 = 1 + 7 + 857 + 93 + 90 + 4 + 8 + 96
= 1 + 78 + 5 + 79 + 3 + 90 + 4 + 896 = 1 + 78 + 57 + 9 + 3 + 904 + 8 + 96
= 1 + 78 + 57 + 93 + 904 + 8 + 9 + 6 = 1 + 78 + 579 + 390 + 4 + 8 + 96
= 17 + 85 + 7 + 939 + 04 + 8 + 96 = 17 + 857 + 93 + 90 + 4 + 89 + 6
= 178 + 5 + 7 + 939 + 04 + 8 + 9 + 6 = 178 + 57 + 9 + 3 + 9 + 04 + 896 = 342.
342 = 1156 appears in the decimal expressions of π and e:
π = 3.14159•••1156••• (from the 6926th digit),
e = 2.71828•••1156••• (from the 12865th digit).
Page of Squares : First Upload November 24, 2003 ; Last Revised January 27, 2011
by Yoshio Mimura, Kobe, Japan