2800
28002 = 63 + 1123 + 1863 = 1003 + 1403 + 1603 = 204 + 404 + 404 + 404.
A cubic polynomial:
(X + 19352)(X + 26882)(X + 28002) = X3 + 43372X2 + 106327202X + 145635840002.
Komachi equations:
28002 = 92 * 82 * 72 / 62 * 52 * 42 / 32 / 22 * 102 = 982 / 72 * 62 * 52 * 42 / 32 / 22 * 102
= 982 / 72 / 62 * 52 * 42 * 32 * 22 * 102.
by Yoshio Mimura, Kobe, Japan
2801
28012 = 7845601, a square with different digits.
A cubic polynomial:
(X + 4482)(X + 6842)(X + 26792) = X3 + 28012X2 + 22118282X + 8209313282.
by Yoshio Mimura, Kobe, Japan
2802
28022 = 7851204, a square with different digits.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2806
2806 = (12 + 22 + 32 + ... + 912) / (12 + 22 + 32 + 42 + 52 + 62).
897k + 2806k + 9890k + 12328k are squares for k = 1,2,3 (1612, 160772, 16922712).
Page of Squares : First Upload November 25, 2008 ; Last Revised May 24, 2011by Yoshio Mimura, Kobe, Japan
2807
28072 = 284 + 324 + 344 + 474.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2808
28082 = 393 + 1543 + 1613.
28082 = 7884864, and 7 * 88 * 4 + 86 * 4 = 788 * 4 - 86 * 4 = 2808.
Page of Squares : First Upload May 14, 2007 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2809
The square of 53.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2812
2812k + 4940k + 8056k + 13433k are squares for k = 1,2,3 (1712, 166632, 17577092).
Page of Squares : First Upload May 24, 2011 ; Last Revised May 24, 2011by Yoshio Mimura, Kobe, Japan
2814
28142 = 7918596, a zigzag square.
28142 = (12 + 5)(142 + 5)(812 + 5) = (32 + 5)(142 + 5)(532 + 5).
Page of Squares : First Upload May 14, 2007 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2816
28162 = 105 + 125 + 145 + 185 + 225.
28162 = (12 + 7)(132 + 7)(752 + 7) = (12 + 7)(22 + 7)(32 + 7)(52 + 7)(132 + 7)
= (12 + 7)(22 + 7)(32 + 7)(752 + 7) = (12 + 7)(22 + 7)(52 + 7)(532 + 7) = (22 + 7)(112 + 7)(752 + 7)
= (22 + 7)(52 + 7)(112 + 7)(132 + 7) = (32 + 7)(132 + 7)(532 + 7) = (32 + 7)(52 + 7)(92 + 7)(132 + 7)
= (32 + 7)(92 + 7)(752 + 7) = (52 + 7)(92 + 7)(532 + 7).
203 + 2816 = 1042, 203 - 2816 = 722.
Page of Squares : First Upload July 28, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2817
28172 = 93 + 743 + 1963.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2819
28192 = 7946761, and 7 + 9 + 467 * 6 + 1 = 2819.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2820
A cubic polynomial:
(X + 28202)(X + 42402)(X + 97472) = X3 + 109972X2 + 510531002X + 1165429296002.
by Yoshio Mimura, Kobe, Japan
2821
28212 = 7958041, a zigzag square with different digits.
2821 = (12 + 22 + 32 + ... + 772) / (12 + 22 + 32 + 42 + 52).
Page of Squares : First Upload May 14, 2007 ; Last Revised November 25, 2008by Yoshio Mimura, Kobe, Japan
2822
28222 = 863 + 1493 + 1593.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2823
A 4x4 perfect magic square consisting of squares whose constant is 2823.
|
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007
by Yoshio Mimura, Kobe, Japan
2824
28242 = 184 + 304 + 304 + 504.
28242 = 7974976, 72 + 92 + 72 + 42 + 92 + 72 + 62 = 192.
Page of Squares : First Upload May 14, 2007 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2825
28252 = 7980625, a square with different digits.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2828
A cubic polynomial:
(X + 1922)(X + 28282)(X + 99992) = X3 + 103932X2 + 283474682X + 54292170242.
by Yoshio Mimura, Kobe, Japan
2829
28292 = 443 + 903 + 1933.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2830
1010k + 1690k + 2570k + 2830k are squares for k = 1,2,3 (902, 43002, 2133002).
28302 = 8008900 appears in the decimal expressions of π:
π = 3.14159•••8008900••• (from the 33641st digit)
(8008900 is the tenth 7-digit square in the expression of π.)
by Yoshio Mimura, Kobe, Japan
2831
1 / 2831 = 0.00035323...., 32 + 532 + 22 + 32 = 2831.
The square root of 2831 is 53.207..., and 53 = 22 + 02 + 72.
28315 = 181844155292323151 : 182 + 182 + 42 + 42 + 12 + 52 + 52 + 22 + 92 + 22 + 322 + 312 + 52 + 12 = 2831.
Page of Squares : First Upload May 14, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
2832
28322 = 8020224, a square with even digits.
28322 = 423 + 673 + 1973 = 284 + 324 + 324 + 484.
Page of Squares : First Upload May 14, 2007 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2835
28352 = (22 + 5)(42 + 5)(102 + 5)(202 + 5).
Komachi equations: 28352 = 92 / 82 * 72 / 62 * 52 * 4322 */ 12.
28352 = 1053 + 1383 + 1623 = 65 + 65 + 95 + 245.
Page of Squares : First Upload July 28, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2836
(28362 -8) = (32 - 8)(42 - 8)(72 - 8)(112 - 8)(152 - 8).
The quadratic polynomial 2836X2 - 18396X + 31689 takes the values 1272, 792, 452, 592, 1032, 1532 at X = 1, 2,..., 6,
Page of Squares : First Upload May 14, 2007 ; Last Revised December 15, 2008by Yoshio Mimura, Kobe, Japan
2838
The square root of 2838 is 53.27...., and 53 = 22 + 72.
28382 = (12 + 2)(162 + 2)(1022 + 2).
Loop of length 10 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2838 - 2228 - 1268 - 4768 - ... - 1480 - 6596 - 13441 - 2838
(Note f(2838) = 282 + 382 = 2228, f(2228) = 222 + 282 = 1268, etc. See 1268)
by Yoshio Mimura, Kobe, Japan
2839
28392 = 8059921, 82 + 02 + 52 + 92 + 92 + 22 + 12 = 162.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2840
Komachi equation: 28402 = 92 * 82 * 72 + 652 * 432 - 212.
Page of Squares : First Upload May 14, 2007 ; Last Revised September 24, 2010by Yoshio Mimura, Kobe, Japan
2843
28432 = 8082649, a zigzag square.
28432 = 1003 + 1133 + 1783.
(28432 + 1) = (22 + 1)(32 + 1)(42 + 1)(62 + 1)(162 + 1).
Page of Squares : First Upload May 14, 2007 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2845
28452 + 28462 + 28472 + ... + 38392 = 38402 + 38412 + 38422 + ... + 44842.
Page of Squares : First Upload September 13, 2011 ; Last Revised September 13, 2011by Yoshio Mimura, Kobe, Japan
2848
28482 = 83 + 483 + 2003 = 743 + 1023 + 1883 = 44 + 284 + 444 + 444.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2849
28492 = 2942 + 2952 + 2962 + 2972 + 2982 + 2992 + 3002 + ... + 3672,
28492 = 8542 + 8552 + 8562 + 8572 + 8582 + 8592 + 8602 + ... + 8642.
by Yoshio Mimura, Kobe, Japan
2850
28502 = 8122500, with 81 = 92 and 22500 = 1502.
28502 = 133 + 713 + 1983.
1482k + 2850k + 32718k + 79914k are squares for k = 1,2,3 (3422, 864122, 233538122).
Komachi equation: 28502 = 92 / 82 * 762 * 52 * 42 / 32 / 22 * 102.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 24, 2011by Yoshio Mimura, Kobe, Japan
2852
28522 = 8133904 appears in the decimal expressions of π:
π = 3.14159•••8133904••• (from the 1984th digit)
(8133904 is the first 7-digit square in the expression of π.)
by Yoshio Mimura, Kobe, Japan
2853
28535 = 189020478521729493 : 12 + 82 + 92 + 02 + 22 + 02 + 42 + 72 + 82 + 52 + 22 + 12 + 72 + 22 + 92 + 492 + 32 = 2853.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
2855
28552 = 3792 + 3802 + 3812 + 3822 + 3832 + 3842 + 3852 + ... + 4282.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2856
28562 = (22 - 1)(332 - 1)(502 - 1).
The integral triangle of sides 801, 21097, 21320 has square area 28562.
Page of Squares : First Upload October 11, 2011 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2857
28572 = 244 + 334 + 344 + 484.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2859
28595 = 191016453191578299 : 12 + 92 + 102 + 162 + 452 + 32 + 12 + 92 + 12 + 52 + 72 + 82 + 22 + 92 + 92 = 2859.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
2863
28632 = 8196769, a zigzag square.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2866
28662 = 8213956, a square with different digits.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2867
28672 = 353 + 813 + 1973.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2868
28682 = 103 + 473 + 2013 = 343 + 963 + 1943.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2870
28702 = 1279 x 1280 + 1281 x 1282 + 1283 x 1284 + 1285 x 1286 + ... + 1287 x 1288.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2872
1 / 2872 = 0.0003481...., and 3481 = 592.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2873
28732 = 24 + 134 + 444 + 464 = 134 + 264 + 264 + 524.
341 + 21341 + 28731 = 712, 342 + 21342 + 28732 = 35792, 343 + 21343 + 28733 = 1828452 (See 71).
Page of Squares : First Upload July 28, 2008 ; Last Revised January 11, 2011by Yoshio Mimura, Kobe, Japan
2874
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2874 - 6260 - 7444 - 7412 - ... - 9333 - 9738 - 10853 - 2874
(Note f(2874) = 282 + 742 = 6260, f(6260) = 622 + 602 = 7444, etc. See 41)
by Yoshio Mimura, Kobe, Japan
2875
28752 = 8265625, a zigzag square.
230k + 610k + 665k + 2720k are squares for k = 1,2,3 (652, 28752, 1437252).
Page of Squares : First Upload May 14, 2007 ; Last Revised May 24, 2011by Yoshio Mimura, Kobe, Japan
2878
28782 = 8282884, a square with 3 kinds of even digits.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2880
28802 = (22 - 1)(32 - 1)(192 - 1)(312 - 1) = (22 - 1)(32 - 1)(42 - 1)(52 - 1)(312 - 1)
= (22 - 1)(32 - 1)(42 - 1)(92 - 1)(172 - 1) = (22 - 1)(52 - 1)(112 - 1)(312 - 1)
= (22 - 1)(52 - 1)(72 - 1)(492 - 1) = (22 - 1)(92 - 1)(112 - 1)(172 - 1)
= (32 - 1)(52 - 1)(112 - 1)(192 - 1) = (42 - 1)(52 - 1)(92 - 1)(172 - 1)
= (52 - 1)(192 - 1)(312 - 1) = (92 - 1)(172 - 1)(192 - 1).
822 + 2880 = 982, 822 - 2880 = 622.
Page of Squares : First Upload July 27, 2011 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2882
2882 = (12 + 22 + 32 + ... + 15722) / (12 + 22 + 32 + ... + 1102).
28822 = 8305924, a square with different digits.
2882k + 26986k + 48601k + 75980k are squares for k = 1,2,3 (3932, 941892, 239395952).
28822 = (54 + 55 + 56 + 57)2 + (58 + 59 + 60 + 61)2 + (62 + 63 + 64 + 65)2 + ... + (182 + 183 + 184 + 185)2.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 24, 2011by Yoshio Mimura, Kobe, Japan
2883
28832 = 8311689, 82 + 32 + 12 + 12 + 62 + 82 + 92 = 162.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2884
28842 = 8317456, a square with different digits.
The square root of 2884 is 53.702...., and 53 = 72 + 02 + 22.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2886
The square root of 2886 is 53.72...., and 53 = 72 + 22.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2888
28882 = 1143 + 1903 = 384 + 384 + 384 + 384.
2888, 2889, 2890, 2891 and 2892 are five consecutive integers having square factors (the third case).
Page of Squares : First Upload July 28, 2008 ; Last Revised December 14, 2013by Yoshio Mimura, Kobe, Japan
2890
28902 = (12 + 1)(42 + 1)(132 + 1)(382 + 1).
28902 = (12 + 1)(42 + 1)(132 + 1)(382 + 1).
578k + 2074k + 2890k + 4862k are squares for k = 1,2,3 (1022, 60522, 3849482).
Page of Squares : First Upload May 24, 2011 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2892
28922 = 1003 + 1463 + 1623.
28922 = 8363664, and 836 * 3 + 6 * 64 = 2892.
Page of Squares : First Upload May 14, 2007 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
2894
28942 = 8375236, 82 + 32 + 72 + 52 + 22 + 32 + 62 = 142.
28942 = 8375236, and 83 * 7 * 5 - 2 - 3 - 6 = 2894.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2896
28962 = 8386816, a zigzag square.
Page of Squares : First Upload May 14, 2007 ; Last Revised May 14, 2007by Yoshio Mimura, Kobe, Japan
2897
28972 = 8392609, a zigzag square.
28973 = 24313388273, and 22 + 432 + 132 + 32 + 82 + 82 + 272 + 32 = 2897.
28972 = 783 + 903 + 1933.
Page of Squares : First Upload May 14, 2007 ; Last Revised December 1, 2008by Yoshio Mimura, Kobe, Japan
2898
28982 = 74 + 294 + 314 + 514.
28982 = (12 + 5)(32 + 5)(82 + 5)(382 + 5) = (12 + 5)(312 + 5)(382 + 5)
= (12 + 5)(42 + 5)(82 + 5)(312 + 5) = (172 + 5)(1692 + 5) = (22 + 5)(32 + 5)(82 + 5)(312 + 5)
= (32 + 5)(42 + 5)(1692 + 5) = (82 + 5)(112 + 5)(312 + 5).
The integral triangle of sides 5993, 6335, 12006 has square area 28982.
Komachi equation: 28982 = 12 * 232 / 42 * 5672 * 82 / 92.
Page of Squares : First Upload July 28, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan