2500
2500 = 50^2.
55 + 2500 = 752, 55 - 2500 = 252.
Page of Squares : First Upload July 27, 2011 ; Last Revised July 27, 2011by Yoshio Mimura, Kobe, Japan
2502
25022 = 6260004, a square with even digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2503
25032 = 1453 + 205 + 47.
Page of Squares : First Upload January 6, 2011 ; Last Revised January 6, 2011by Yoshio Mimura, Kobe, Japan
2504
25042 = 24 + 104 + 104 + 504.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2505
25052 = 813 + 943 + 1703.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2506
25062± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
2507
25072 = 6285049, a square with different digits.
667k + 2507k + 2553k + 2737k are squares for k = 1,2,3 (922, 45542, 2306442).
Page of Squares : First Upload April 23, 2007 ; Last Revised May 20, 2011by Yoshio Mimura, Kobe, Japan
2508
25082 = (12 + 8)(62 + 8)(1262 + 8) = (22 + 8)(52 + 8)(1262 + 8) = (62 + 8)(72 + 8)(502 + 8)
= (72 - 1)(3622 - 1).
1122k + 1353k + 2508k + 4818k are squares for k = 1,2,3 (992, 57092, 3626372).
Page of Squares : First Upload May 20, 2011 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2509
25092 = 6295081, a square with different digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2510
25102± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
2512
25122 = 603 + 643 + 1803.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2515
25152 = 6325225, and 63 * 2 * 5 * 2 * 2 - 5 = 2515.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2516
25162 = (82 + 4)(122 + 4)(252 + 4).
Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2518
25182 = 6340324, a zigzag square.
25182 = 6340324, and 63 * 40 - 3 * 2 + 4 = 2518.
The square root of 2518 is 50.17... with 50 = 12 + 72.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2519
1 / 2519 = 0.0003969..., and 3969 = 632.
25192 = 6345361, and 63 * 4 * 5 / 3 * 6 - 1 = 2519.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2520
532 + 2520 = 732, 532 - 2520 = 172.
25202 = (22 - 1)(32 - 1)(62 - 1)(82 - 1)(112 - 1) = (22 - 1)(42 - 1)(132 - 1)(292 - 1)
= (22 - 1)(42 - 1)(52 - 1)(62 - 1)(132 - 1) = (22 - 1)(62 - 1)(132 - 1)(192 - 1)
= (22 - 1)(62 - 1)(82 - 1)(312 - 1) = (262 - 1)(972 - 1) = (32 - 1)(42 - 1)(52 - 1)(62 - 1)(82 - 1)
= (32 - 1)(42 - 1)(82 - 1)(292 - 1) = (32 - 1)(62 - 1)(82 - 1)(192 - 1)
= (52 - 1)(62 - 1)(82 - 1)(112 - 1) = (82 - 1)(112 - 1)(292 - 1).
Komachi equations:
25202 = 12 * 22 / 32 / 42 * 52 * 62 * 72 * 82 * 92 = 12 / 22 * 32 * 42 * 52 / 62 * 72 * 82 * 92
= 12 * 22 / 32 * 452 / 62 * 72 * 82 * 92 = 12 / 22 * 32 * 452 * 62 * 72 * 82 / 92
= 12 * 23452 / 672 * 82 * 92 = 92 * 82 * 72 * 62 * 52 / 42 / 32 * 22 */ 12
= 92 * 82 * 72 / 62 * 52 * 42 * 32 / 22 */ 12 = 982 / 72 * 62 * 52 * 42 * 32 / 22 */ 12.
by Yoshio Mimura, Kobe, Japan
2521
25212 = 813 + 1203 + 1603.
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2521 - 1066 - 4456 - 5072 - ... - 7412 - 5620 - 3536 - 2521
(Note f(2521) = 252 + 212 = 1066, f(1066) = 102 + 662 = 4456, etc. See 41)
by Yoshio Mimura, Kobe, Japan
2524
The quadratic polynomial 2524X2 - 10388X + 13489 takes the values 752, 532, 712, 1112, 1572, 2052 at X = 1, 2,..., 6,
Page of Squares : First Upload December 15, 2008 ; Last Revised December 15, 2008by Yoshio Mimura, Kobe, Japan
2525
25252 = 6375625, a zigzag square.
25252 = 893 + 1083 + 1643.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2526
12 + 22 + 32 + 42 + ... + 25262 = 5375719951, which consists of odd digits,
the second 10-digit sum (there are 4 10-digit sum in all.)
by Yoshio Mimura, Kobe, Japan
2527
25272 = 6385729, a zigzag square with different digits.
25272 = 194 + 384 + 384 + 384.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2528
25282 = 6390784, a square with different digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2529
25292 = 6395841, a square with different digits.
25292 = 63 + 403 + 1853.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2530
25302 = 6400900, and 6400 = 802, 900 = 302.
25302= 516 x 517 + 517 x 518 + 518 x 519 + 519 x 520 + ... + 538 x 539.
25302 = 2252 + 2262 + 2272 + 2282 + 2292 + 2302 + ... + 3122.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2532
S2(2532) = S2(1569) + S2(2315), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2534
25342 = 353 + 363 + 1853.
Komachi equations: 25342 = 982 / 72 / 62 * 5432 * 22 */ 12.
25344 = 41231244376336, and 412 + 232 + 122 + 42 + 42 + 32 + 72 + 62 + 32 + 32 + 62 = 2534.
Page of Squares : First Upload July 24, 2008 ; Last Revised September 21, 2010by Yoshio Mimura, Kobe, Japan
2535
25352 = 783 + 1043 + 1693.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2537
25372 = 82 + 9512 + 23522 = 25322 + 1592 + 82.
25372 = 30 + 33 + 310 + 313 + 314.
Page of Squares : First Upload August 29, 2011 ; Last Revised September 7, 2013by Yoshio Mimura, Kobe, Japan
2538
25382 = 6441444, a square with just 3 kinds of digits.
25382 = 74 + 214 + 414 + 434.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2541
25412 = 12 + 142 + 272 + ... + (13x+1)2 + ... + 6252.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2544
25442 = 6471936, a zigzag square.
25442 = 6471936, and 6 + 4 * 71 * 9 - 3 * 6 = 2544.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2545
(25452 + 5) = (42 + 5)(52 + 5)(82 + 5)(122 + 5) = (12 + 5)(82 + 5)(102 + 5)(122 + 5).
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2547
25472 = 6487209, a square with different digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2548
25482 = 6492304, a zigzag square.
Komachi equation: 25482 = 982 * 72 * 652 * 42 * 32 / 2102.
Page of Squares : First Upload April 23, 2007 ; Last Revised September 21, 2010by Yoshio Mimura, Kobe, Japan
2550
25502 = (52 + 9)(92 + 9)(462 + 9).
2550k + 2652k + 6681k + 11526k are squares for k = 1,2,3 (1532, 138212, 13655252).
Page of Squares : First Upload May 20, 2011 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2552
25522 = 6512704, a square with different digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2553
25532 = 6517809, a square with different digits.
25532 = 6517809, 62 + 52 + 12 + 72 + 82 + 02 + 92 = 162.
667k + 2507k + 2553k + 2737k are squares for k = 1,2,3 (922, 45542, 2306442).
Page of Squares : First Upload April 23, 2007 ; Last Revised May 20, 2011by Yoshio Mimura, Kobe, Japan
2556
1 / 2556 = 0.0003912363067..., 392 + 12 + 22 + 32 + 62 + 302 + 62 + 72 = 2556.
25562 + 25572 + 25582 + ... + 78272 = 78282 + 78292 + 78302 + ... + 98042.
The square root of 2556 is 50.55...., 50 = 52 + 52.
Page of Squares : First Upload April 23, 2007 ; Last Revised September 13, 2011by Yoshio Mimura, Kobe, Japan
2557
25572 = 6538249, a square with different digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2558
Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2558 - 3989 - 9442 - 10600 - ... - 693 - 8685 - 14621 - 2558
(Note f(2558) = 252 + 582 = 3989, f(3989) = 392 + 892 = 9442, etc. See 37)
by Yoshio Mimura, Kobe, Japan
2559
25592 = 6548481, and 6 * 54 * 8 - 4 * 8 - 1 = 2559.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2564
25642 = 593 + 763 + 1813 = 84 + 204 + 204 + 504.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2565
Komachi equations: 25652 = 92 / 82 * 762 * 52 * 42 * 32 / 22 */ 12.
Page of Squares : First Upload July 24, 2008 ; Last Revised September 21, 2010by Yoshio Mimura, Kobe, Japan
2566
25662± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
2568
25682 = 6594624, a zigzag square.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2569
25692 = 1733 + 175 + 37.
A cubic polynomial :
(X + 6082)(X + 13112)(X + 21242) = X3 + 25692X2 + 31712522X + 16930149122.
25692 = 6599761 appears in the decimal expressions of e:
e = 2.71828•••6599761••• (from the 143717th digit)
by Yoshio Mimura, Kobe, Japan
2570
1010k + 1690k + 2570k + 2830k are squares for k = 1,2,3 (902, 43002, 2133002).
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2570 - 5525 - 3650 - 3796 - ... - 1681 - 6817 - 4913 - 2570
(Note f(2570) = 252 + 702 = 5525, f(5525) = 552 + 252 = 3650, etc. See 41)
by Yoshio Mimura, Kobe, Japan
2572
The square root of 2572 is 50.71..., and 50 = 72 + 12.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2573
25732 = 30 + 32 + 33 + 35 + 38 + 310 + 311 + 313 + 314.
Page of Squares : First Upload August 29, 2011 ; Last Revised August 29, 2011by Yoshio Mimura, Kobe, Japan
2575
25752 = 104 + 354 + 404 + 404.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2577
25772 = 84 + 234 + 324 + 484.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2578
25782 = 6646084, a square with even digits.
25782 = 33 + 683 + 1853.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2580
25802 = 6656400, and 6 * 6 * 5 + 6 * 400 = 2580.
25802 = (21 + 22 + 23 + ... + 40)2 + (41 + 42 + 43 + ... + 60)2 + (61 + 62 + 63 + ... + 80)2 + ... + (81 + 82 + 83 + ... + 100)2.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2581
25812 = 6661561, a square with just 3 kinds of digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2583
25832 = (42 + 5)(62 + 5)(882 + 5).
Cubic polynomials :
(X + 14162)(X + 15682)(X + 25832) = X3 + 33372X2 + 58915922X + 57350039042,
(X + 18042)(X + 21122)(X + 25832) = X3 + 37932X2 + 81234122X + 98413539842.
by Yoshio Mimura, Kobe, Japan
2586
25862 = S2(209) + S2(221), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan
2589
1 / 2589 = 0.000386249517..., 32 + 82 + 62 + 22 + 492 + 52 + 12 + 72 = 2589.
25892± 2 are primes.
Page of Squares : First Upload April 23, 2007 ; Last Revised December 29, 2013by Yoshio Mimura, Kobe, Japan
2591
25912 = 6713281, a zigzag square.
25912 = 6713281, and 6 - 7 * 1 + 32 * 81 = 6 - 7 / 1 + 32 * 81 = 6 - 7 + 1 * 32 * 81 = 2591.
25912 = 703 + 1303 + 1613.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2592
25922 = 6718464, a zigzag square.
Komachi equation: 25922 = 92 * 82 * 72 * 62 * 542 / 32 / 212.
25922 = 123 + 963 + 1803 = 543 + 903 + 1803 = 364 + 364 + 364 + 364 = 68 + 68 + 68 + 68.
Page of Squares : First Upload April 23, 2007 ; Last Revised September 21, 2010by Yoshio Mimura, Kobe, Japan
2593
25932 = 6723649, and 6 + 72 * 36 + 4 - 9 = 2593.
25932 = 284 + 284 + 284 + 474.
Page of Squares : First Upload April 23, 2007 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2595
25952 = 6734025, a square with different digits.
25952 = 473 + 1453 + 1533.
25952 = 6734025 appears in the decimal expressions of π:
π = 3.14159•••6734025••• (from the 39300th digit)
by Yoshio Mimura, Kobe, Japan
2596
25962 = 2162 + 2172 + 2182 + 2192 + 2202 + 2212 + ... + 3112.
154k + 814k + 1298k + 2090k are squares for k = 1,2,3 (662, 25962, 1089002).
Page of Squares : First Upload April 23, 2007 ; Last Revised May 20, 2011by Yoshio Mimura, Kobe, Japan
2598
25982 = 433 + 1043 + 1773.
Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008by Yoshio Mimura, Kobe, Japan
2599
25992 = 6754801, a square with different digits.
Page of Squares : First Upload April 23, 2007 ; Last Revised April 23, 2007by Yoshio Mimura, Kobe, Japan