The smallest squares containing k 44's :
144 = 122, 44944 = 2122, 614444944 = 247882,
447544944144 = 6689882, 444431244445444 = 210815382.
The squares which begin with 44 and end in 44 are
44944 = 2122, 4460544 = 21122, 44063044 = 66382, 44382244 = 66622,
44729344 = 66882,...
44 is the sum of m squares for m = 3, 4, ..., 30.
The 3rd integer which is the sum of 6 squares in just 4 ways:
[1,1,1,1,2,6], [1,1,1,3,4,4], [1,1,2,2,3,5], [2,2,3,3,3,3].
442 = 1936 is a zigzag square, whose digits are distinct.
442 = (22 + 7)(132 + 7).
442 = 1936, a square with odd digits except the last digit.
442 = 1936, 1999396 = 14142.
442 = 1936, 1 = 12, 9 = 32, 36 = 62.
442 = 1936, 1 + 93 + 6 = 102.
44944 = 2122.
44 and 45 are consecutive integers having square factors (the 4th case).
53 + 44 = 132, 53 - 44 = 92.
A cubic polynomial : (X + 442)(X + 572)(X + 1442) = X3 + 1612X2 + 106682X + 3611522.
10k + 44k + 57k + 58k are squares for k = 1,2,3 (132, 932, 6832).
Komachi equations:
442 = 98 + 7 * 65 * 4 - 3 + 21 = 9 - 8 * 7 + 654 * 3 + 21
= - 9 + 8 - 7 + 6 * 54 * 3 * 2 * 1, and more 3 equations,
442 = 9 - 8 - 7 + 654 * 3 - 2 * 10 = 98 * 7 - 6 * 5 + 4 * 32 * 10
= - 98 / 7 + 654 * 3 - 2 - 10,
442 = 12 * 22 * 32 + 42 - 52 + 62 * 72 + 82 + 92 = 12 * 22 + 32 + 452 - 62 - 72 + 82 - 92
= 12 * 232 - 452 - 672 + 892,
442 = 92 + 82 + 72 * 62 + 52 + 42 - 32 - 22 - 12 = 92 + 82 + 72 * 62 - 52 + 42 + 32 * 22 * 12
= 92 + 82 + 72 * 62 - 52 + 42 + 32 * 22 / 12 = 982 - 762 - 542 + 322 * 12
= 982 - 762 - 542 + 322 / 12,
442 = 13 - 23 + 33 * 43 + 53 + 63 - 73 - 83 + 93 = - 13 - 23 + 33 * 43 - 53 + 63 + 73 + 83 - 93,
442 = 93 - 83 - 73 + 63 + 53 + 43 * 33 - 23 + 13 = - 93 + 83 + 73 + 63 - 53 + 43 * 33 - 23 - 13.
442 = 2 x 3 + 3 x 4 + 4 x 5 + ... + 17 x 18.
(442 + 2) = (62 + 2)(72 + 2).
12 + 22 + 32 + 42 + ... + 442 = 29370, which consists of different digits.
442 + 452 + 462 + ... + 672 = 2742,
442 + 452 + 462 + ... + 932 = 4952.
(1 + 2 + ... + 4)(5 + 6 + ... + 19)(20 + 21 + ... + 44) = 12002,
(1 + 2 + ... + 6)(7 + 8 + ... + 27)(28 + 29 + ... + 44) = 21422,
(1 + 2 + ... + 9)(10 + 11 + ... + 30)(31 + 32 + ... + 44) = 31502,
(1 + 2 + ... + 16)(17 + 18 + ... + 27)(28 + 29 + ... + 44) = 44882,
(1 + 2 + ... + 17)(18 + 19 + ... + 27)(28 + 29 + ... + 44) = 45902.
(13 + 23 + ... + 203)(213 + 223 + ... + 353)(363 + 373 + ... + 443) = 952560002.
Page of Squares : First Upload December 29, 2003 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan