5000
50002 = 504 + 504 + 504 + 504
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5001
50012 = 25010001, 2 + 50 * 100 + 0 - 1 = 2 + 50 * 100 - 1 = 5001.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5002
50022 = 25020004, 22 + 52 + 02 + 22 + 02 + 02 + 02 + 42 = 72.
50022 = (92 + 1)(112 + 1)(502 + 1).
Page of Squares : First Upload October 22, 2007 ; Last Revised November 2, 2013by Yoshio Mimura, Kobe, Japan
5003
50032 = 403 + 413 + 2923
50035 = 3134386256752025243 : 32 + 132 + 42 + 32 + 82 + 62 + 22 + 52 + 62 + 72 + 52 + 22 + 02 + 22 + 522 + 432 = 32 + 132 + 42 + 32 + 82 + 62 + 22 + 52 + 62 + 72 + 522 + 02 + 22 + 52 + 22 + 432 = 32 + 132 + 432 + 82 + 62 + 22 + 52 + 62 + 72 + 52 + 22 + 02 + 22 + 522 + 42 + 32 = 32 + 132 + 432 + 82 + 62 + 22 + 52 + 62 + 72 + 522 + 02 + 22 + 52 + 22 + 42 + 32 = 5003.
Page of Squares : First Upload September 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
5004
50042 = 723 + 1573 + 2753
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5005
5005 = (12 + 22 + 32 + ... + 14622) / (12 + 22 + 32 + ... + 852).
50052 = 25050025, a square with just 3 kinds of digits.
50052 = 25050025, 2 * 50 * 50 + 0 * 2 + 5 = 5005.
50052 = (62 - 1)(8462 - 1).
5005k + 17017k + 23881k + 35893k are squares for k = 1,2,3 (2862, 466182, 80569062).
Page of Squares : First Upload October 22, 2007 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
5006
50062± 3 are primes.
Page of Squares : First Upload November 4, 2008 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
5007
50072 = 25070049 appears in the decimal expressions of π:
π = 3.14159•••25070049••• (from the 50638th digit)
(25070049 is the ninth 8-digit square in the expression of π.)
by Yoshio Mimura, Kobe, Japan
5012
50122 = 25120144, 251 * 20 - 1 * 4 - 4 = 251 * 20 * 1 - 4 - 4 = 5012.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5015
5015 = (12 + 22 + 32 + ... + 592) / (12 + 22 + 32).
50152 = 25150225, 2 - 5 + 1 + 5022 - 5 = 5015.
50155 = 3172157095016384375 : 312 + 72 + 212 + 572 + 02 + 92 + 52 + 02 + 12 + 62 + 32 + 82 + 42 + 32 + 72 + 52 = 5015.
Page of Squares : First Upload October 22, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
5016
50162 = (32 - 1)(52 - 1)(3622 - 1) = (42 + 8)(122 + 8)(832 + 8).
969k + 5016k + 5092k + 6612k are squares for k = 1,2,3 (1332, 97852, 7404112).
Page of Squares : First Upload June 17, 2011 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
5017
50172 = 893 + 1713 + 2693 = 74 + 144 + 444 + 684
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5018
50182 = 25180324, a zigzag square.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5023
50232 = 25230529, a zigzag square.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5025
50252 = 1203 + 1703 + 2653
50252 = 25250625, a zigzag square.
Page of Squares : First Upload October 22, 2007 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5027
50272 = 25270729, a zigzag square.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5028
50282 = 44 + 44 + 544 + 644
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5029
50294 = 639626638487281, and 62 + 32 + 92 + 622 + 62 + 62 + 32 + 82 + 42 + 82 + 72 + 282 + 12 = 5029.
Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008by Yoshio Mimura, Kobe, Japan
5030
4-cycle : 50302 = 25300900 -- 0092 = 09054081 -- 05402 = 00291600 -- 29162 = 08503056,
Other examples : 1600 -- 5600 -- 3600 -- 9600 -- 1600, 2100 -- 4100 -- 8100 -- 6100 -- 2100.
by Yoshio Mimura, Kobe, Japan
5031
50312± 2 are primes.
Page of Squares : First Upload December 29, 2013 ; Last Revised December 29, 2013by Yoshio Mimura, Kobe, Japan
5032
50322 = 1402 + 1412 + 1422 + 1432 + 1442 + ... + 4282.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5036
50362 = 25361296, 22 + 52 + 32 + 62 + 12 + 22 + 92 + 62 = 142.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5037
50374 = 643706364934161, and 62 + 42 + 32 + 72 + 02 + 62 + 362 + 492 + 342 + 12 + 62 + 12 = 5037.
Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008by Yoshio Mimura, Kobe, Japan
5039
50392 = 153 + 2253 + 2413
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5040
50402 = 10! + 10! + 10! + 10! + 10! + 10! + 10!.
50402 = (132 - 1)(152 - 1)(262 - 1) = (22 - 1)(42 - 1)(62 - 1)(1272 - 1) = (22 - 1)(412 - 1)(712 - 1)
= (22 - 1)(62 - 1)(172 - 1)(292 - 1) = (22 - 1)(62 - 1)(72 - 1)(712 - 1)
= (22 - 1)(62 - 1)(72 - 1)(82 - 1)(92 - 1) = (22 - 1)(62 - 1)(92 - 1)(552 - 1)
= (22 - 1)(82 - 1)(92 - 1)(412 - 1) = (42 - 1)(62 - 1)(132 - 1)(172 - 1)
= (42 - 1)(82 - 1)(112 - 1)(152 - 1) = (82 - 1)(132 - 1)(492 - 1) = (82 - 1)(92 - 1)(712 - 1).
The integral triangle of sides 5128, 68625, 73703 has square area 50402.
Komachi equations:
50402 = 982 / 72 / 62 * 52 * 4322 */ 12.
by Yoshio Mimura, Kobe, Japan
5041
The square of 71.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5044
50442 = 25441936, 2 * 5 + 4 * 419 * 3 + 6 = 5044.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5046
50462± 5 are primes.
50462 = 1323 + 1913 + 2533
50462 = 25462116, 2 * 5 * 4 * 6 * 21 + 1 * 6 = 2 * 5 * 4 * 6 * 21 * 1 + 6 = 5046.
5046, 5047, 5048, 5049 and 5050 are five consecutive integers having square factors (the 5th case).
Page of Squares : First Upload November 10, 2008 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
5047
50472 = 144 + 494 + 564 + 564
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5048
50482 = 25482304, a zigzag square.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5049
50492 = 25492401, a zigzag square.
50492 = (12 + 8)(32 + 8)(52 + 8)(712 + 8) = (32 + 8)(172 + 8)(712 + 8)
= (192 + 2)(2652 + 2) = (32 + 2)(52 + 2)(72 + 2)(412 + 2) = (72 + 2)(7072 + 2).
The square root of 5049 is 71.05631..., and 71 = 02 + 52 + 62 + 32 + 12.
Page of Squares : First Upload October 22, 2007 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
5050
50502 = 25502500, 25 + 5025 + 0 + 0 = 25 + 5025 + 0 * 0 = 2550 * 2 - 50 + 0 = 2550 + 2500 = 5050.
50502 + 50512 + 50522 + ... + 51002 = 51012 + 51022 + 51032 + ... + 51502.
Page of Squares : First Upload October 22, 2007 ; Last Revised September 16, 2011by Yoshio Mimura, Kobe, Japan
5052
50522 = 383 + 2223 + 2443
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5054
50542 = 25542916, 2554 * 2 - 9 * 1 * 6 = 5054.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5055
50552 = 683 + 2143 + 2493
50552 = S2(329) + S2(344), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.
Page of Squares : First Upload October 22, 2007 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5057
50572 = 563 + 1693 + 2743
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5058
50582 = 13 + 983 + 2913
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5059
50592 = 25593481, 22 + 52 + 52 + 92 + 32 + 42 + 82 + 12 = 152.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5060
50602± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
5062
50625 = 3323614952324532832 : 32 + 322 + 362 + 142 + 92 + 52 + 22 + 32 + 242 + 52 + 32 + 282 + 322 = 5062.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
5063
50632 is the sum of consecutive primes : 2, 3, 5, 7, 11,.., 22073.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5065
Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
5065 - 6725 - 5114 - 2797 - ... - 1090 - 8200 - 6724 - 5065
(Note f(5065) = 502 + 652 = 6725, f(6725) = 672 + 252 = 5114, etc. See 37)
by Yoshio Mimura, Kobe, Japan
5068
50682± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
5071
50712 = 25715041, 22 + 52 + 72 + 12 + 52 + 02 + 42 + 12 = 112.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5072
50722 = 25725184, 2 - 57 * 2 + 5184 = 5072.
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
5072 - 7684 - 12832 - ... - 2521 - 1066 - 4456 - 5072
(Note f(5072) = 502 + 722 = 7684, f(7684) = 762 + 842 = 12832, etc. See 41)
by Yoshio Mimura, Kobe, Japan
5074
5074 = (12 + 22 + 32 + ... + 4722) / (12 + 22 + 32 + ... + 272).
Page of Squares : First Upload November 25, 2008 ; Last Revised November 25, 2008by Yoshio Mimura, Kobe, Japan
5076
50762 = 1263 + 1983 + 2523 = 1443 + 1713 + 2613
Komachi equation: 50762 = 9872 * 62 * 542 / 32 / 212.
Page of Squares : First Upload September 8, 2008 ; Last Revised October 19, 2010by Yoshio Mimura, Kobe, Japan
5077
1 / 5077 = 0.000196..., and 196 = 142. Also 5078, 5079,..., 5102.
50775 = 3373151264782809157 : 32 + 32 + 72 + 32 + 152 + 122 + 642 + 72 + 82 + 22 + 82 + 02 + 92 + 152 + 72 = 5077.
Page of Squares : First Upload October 22, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
5080
The square root of 5080 is 71.27411..., and 71 = 22 + 72 + 42 + 12 + 12.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5081
(50812 - 7) = (62 - 7)(82 - 7)(112 - 7)(122 - 7).
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5083
50832 = 1322 + 1332 + 1342 + 1352 + 1362 + ... + 4302.
Page of Squares : First Upload October 22, 2007 ; Last Revised October 22, 2007by Yoshio Mimura, Kobe, Japan
5084
50842 = 1033 + 2093 + 2503
682k + 5084k + 20553k + 20770k are squares for k = 1,2,3 (2172, 296672, 42159072).
Page of Squares : First Upload September 8, 2008 ; Last Revised June 17, 2011by Yoshio Mimura, Kobe, Japan
5085
Komachi equations:
50852 = 12 * 22 / 32 / 42 * 52 * 6782 * 92 = 12 / 22 * 32 * 452 * 6782 / 92.
by Yoshio Mimura, Kobe, Japan
5086
50864 = 669122175820816, and 662 + 92 + 12 + 22 + 22 + 12 + 72 + 52 + 82 + 202 + 82 + 12 + 62 = 5086.
Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008by Yoshio Mimura, Kobe, Japan
5087
50872 = 84 + 114 + 464 + 684
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5088
50882 = 493 + 713 + 2943 = 763 + 1163 + 2883 = 883 + 1323 + 2843
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5089
The square root of 5089 is 71.3372..., and 71 = 32 + 32 + 72 + 22.
50895 = 3413204066390084449 : 32 + 412 + 32 + 22 + 02 + 402 + 62 + 62 + 392 + 02 + 02 + 82 + 42 + 42 + 42 + 92 = 5089.
Page of Squares : First Upload October 22, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
5092
50922 = 84 + 164 + 464 + 684
50925 = 3423276469573215232 : 32 + 42 + 232 + 22 + 72 + 62 + 42 + 62 + 92 + 572 + 32 + 22 + 12 + 52 + 22 + 322 = 32 + 42 + 22 + 32 + 22 + 72 + 62 + 42 + 62 + 92 + 572 + 322 + 12 + 52 + 232 + 22 = 32 + 42 + 22 + 322 + 72 + 62 + 42 + 62 + 92 + 572 + 32 + 22 + 12 + 52 + 232 + 22 = 32 + 42 + 232 + 22 + 72 + 62 + 42 + 62 + 92 + 572 + 322 + 12 + 52 + 22 + 32 + 22 = 5092.
969k + 5016k + 5092k + 6612k are squares for k = 1,2,3 (1332, 97852, 7404112).
Page of Squares : First Upload September 8, 2008 ; Last Revised June 17, 2011by Yoshio Mimura, Kobe, Japan
5095
50952 = 93 + 1023 + 2923 = 163 + 1453 + 2843 = 1023 + 2013 + 2563
Page of Squares : First Upload September 8, 2008 ; Last Revised September 8, 2008by Yoshio Mimura, Kobe, Japan
5096
50962 = (72 + 3)(192 + 3)(372 + 3).
Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan