The smallest squares containing k 41's :
441 = 21²,   4116841 = 2029²,   741418441 = 27229²,
241994141041 = 491929²,   4104174186141241 = 64063829².

The squares which begin with 41 and end in 41 are
4116841 = 2029²,   41229241 = 6421²,   41332041 = 6429²,   41873841 = 6471²,
41977441 = 6479²,...

The 8th integer which is the sum of 3 distinct squares: 1² + 2² + 6².
The 4th integer which is the sum of 3 squares in just 2 ways: 1² + 2² + 6² = 3² + 4² + 4².
The 3rd integer which is the sum of 5 squares in just 2 ways:
  1² + 2² + 2² + 4² + 4² = 2² + 2² + 2² + 2² + 5².
The 2nd integer which is the sum of 6 squares in just 4 ways:
  1² + 1² + 1² + 1² + 1² + 6² = 1² + 1² + 1² + 2² + 3² + 5²
  = 1² + 2² + 3² + 3² + 3² + 3² = 2² + 2² + 2² + 2² + 3² + 4².
The 3rd integer which is the sum of 8 squares in just 4 ways (see 40).

41² = 1681, 16 = 4² and 81 = 9².

4116841 = 2029².

Loop of length 56 by the function f(N) = ... + c² + b² + a² where N = ... + 100²c + 100b + a:
41 - 1681 - 6817 - 4913 - 2570 - 5525 - 3650 - 3796 - 10585 - 7251 - 7785 - 13154 - 3878 - 7528 - 6409 - 4177 - 7610 - 5876 - 9140 - 9881 - 16165 - 7947 - 8450 - 9556 - 12161 - 4163 - 5650 - 5636 - 4432 - 2960 - 4441 - 3617 - 1585 - 7450 - 7976 - 12017 - 690 - 8136 - 7857 - 9333 - 9738 - 10853 - 2874 - 6260 - 7444 - 7412 - 5620 - 3536 - 2521 - 1066 - 4456 - 5072 - 7684 - 12832 - 1809 - 405 - 41
(Note f(41) = 41² = 1681,   f(1681) = 16² + 81² = 6817, etc. The longest loop)

82^k + 298^k + 628^k + 673^k are squares for k = 1,2,3 (41², 971², 24073²).
210^k + 241^k + 538^k + 692^k are squares for k = 1,2,3 (41²,933², 22591²).

Komachi Fractions : (41/16)² = 45387/6912, (41/2)² = 574902/1368.

Komachi equations:
41² = 1 + 234 * 5 + 6 + 7 * 8 * 9 = 1 / 2 * 3456 - 7 * 8 + 9
  = 1 + 23 + 4 * 56 * 7 + 89, and more 7 equations,
41² = 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 11 equations,
41² = 9 + 8 * 7 * 6 * 5 + 4 * 3 - 2 * 10 = 9 + 8 * 7 * 6 * 5 - 4 + 3 * 2 - 10,
41² = 98² / 7² + 6² * 5² + 4² * 3² + 21².

41² = 1681, 1 + 6 + 8 + 1 = 4²,
41² = 1681, 16 + 8 + 1 = 5²,
41² = 1681, 168 + 1 = 13².

41² - 1 = 2(5! + 6!)

The altenating sum of the first 41 cubes is a square : 1³ - 2³ + 3³ - ... + 41³ = 189².

(41² - 1) = (6² - 1)(7² - 1),   (41² + 9) = (2² + 9)(11² + 9),
(41² + 2) = (1² + 2)(3² + 2)(7² + 2),
(41² - 7) = (5² - 7)(10² - 7) = (3² - 7)(4² - 7)(10² - 7).

(1 + 2 + ... + 12)(13 + 14 + ... + 39)(40 + 41) = 2106²,
(1 + 2 + ... + 18)(19 + 20 + ... + 38)(39 + 40 + 41) = 3420².

(1² + 2² + ... + 13²)(14² + 15² + ... + 38²)(39² + 40² + 41²) = 267540².

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Page of Squares : First Upload December 15, 2003 ; Last Revised January 27, 2011
by Yoshio Mimura, Kobe, Japan