24000
12 + 22 + 32 + 42 + ... + 240002 = 4608288004000, which consists of even digits.
1702 + 24000 = 2302, 1702 - 24000 = 702.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24009
240092 = 576432081 is a square with different digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24010
240102± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24018
240182± 5 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24020
240202 = 576960400, and 576 = 242, 960400 = 9802.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24024
240242 = (122 - 1)(132 - 1)(1552 - 1) = (22 - 1)(122 - 1)(272 - 1)(432 - 1).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24035
240352 = 577681225, and 5776 = 762, 81225 = 2852.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24037
242372 = 587432169 is a square with different digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24050
240502 = (12 + 1)(52 + 1)(62 + 1)(82 + 1)(682 + 1)
= (12 + 1)(82 + 1)(182 + 1)(1172 + 1)
= (12 + 1)(82 + 1)(312 + 1)(682 + 1)
= (182 + 1)(312 + 1)(432 + 1) = (22 + 1)(32 + 1)(62 + 1)(182 + 1)(312 + 1)
= (22 + 1)(52 + 1)(182 + 1)(1172 + 1) = (22 + 1)(52 + 1)(312 + 1)(682 + 1)
= (22 + 1)(52 + 1)(62 + 1)(82 + 1)(432 + 1) = (22 + 1)(62 + 1)(312 + 1)(572 + 1)
= (22 + 1)(62 + 1)(72 + 1)(82 + 1)(312 + 1) = (22 + 1)(82 + 1)(312 + 1)(432 + 1)
= (52 + 1)(62 + 1)(182 + 1)(432 + 1) = (62 + 1)(72 + 1)(182 + 1)(312 + 1)
= (32 + 4)(112 + 4)(122 + 4)(492 + 4) = (32 + 4)(492 + 4)(1362 + 4).
by Yoshio Mimura, Kobe, Japan
24090
240902 = (182 + 6)(332 + 6)(402 + 6) = (32 + 6)(42 + 6)(332 + 6)(402 + 6).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24154
241542± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24188
241882± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24192
241922 = (22 - 1)(152 - 1)(172 - 1)(552 - 1) = (22 - 1)(32 - 1)(52 - 1)(82 - 1)(1272 - 1)
= (22 - 1)(32 - 1)(72 - 1)(132 - 1)(552 - 1) = (22 - 1)(72 - 1)(82 - 1)(152 - 1)(172 - 1)
= (32 - 1)(52 - 1)(82 - 1)(132 - 1)(172 - 1) = (52 - 1)(72 - 1)(132 - 1)(552 - 1).
by Yoshio Mimura, Kobe, Japan
24206
242062 = (22 + 3)(42 + 3)(192 + 3)(1102 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24210
242102 = (32 + 9)(232 + 9)(2462 + 9).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24242
242422± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24276
242762 = 589324176 is a square with different digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24288
242882 = (232 - 1)(10572 - 1).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24335
243352 = S2(944) + S2(977), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24350
243502 = 592922500, and 5929 = 772, 22500 = 1502.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24354
243542 = 593117316, a square with odd digits except the last digit 6.
243542 = (12 + 2)(22 + 2)(32 + 2)(52 + 2)(112 + 2)(302 + 2)
= (12 + 2)(22 + 2)(42 + 2)(192 + 2)(712 + 2) = (12 + 2)(32 + 2)(42 + 2)(142 + 2)(712 + 2)
= (12 + 2)(52 + 2)(82 + 2)(112 + 2)(302 + 2) = (22 + 2)(32 + 2)(52 + 2)(112 + 2)(522 + 2)
= (22 + 2)(32 + 2)(52 + 2)(82 + 2)(712 + 2) = (22 + 2)(712 + 2)(1402 + 2)
= (32 + 2)(112 + 2)(222 + 2)(302 + 2) = (32 + 2)(42 + 2)(52 + 2)(112 + 2)(302 + 2)
= (52 + 2)(112 + 2)(142 + 2)(302 + 2) = (52 + 2)(82 + 2)(112 + 2)(522 + 2).
by Yoshio Mimura, Kobe, Japan
24360
243602 = (32 - 1)(592 - 1)(1462 - 1).
4212 + 24360 = 4492, 4212 - 24360 = 3912.
Page of Squares : First Upload March 31, 2012 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24381
243812 = (22 + 5)(42 + 5)(342 + 5)(522 + 5).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24388
243882± 3 are primes.
243882 = (22 + 3)(82 + 3)(192 + 3)(592 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24400
244002 = (12 + 4)(22 + 4)(42 + 4)(222 + 4)(392 + 4) = (22 + 4)(62 + 4)(13642 + 4)
= (42 + 4)(62 + 4)(222 + 4)(392 + 4).
by Yoshio Mimura, Kobe, Japan
24402
244022 = (172 + 5)(242 + 5)(592 + 5) = (32 + 5)(42 + 5)(242 + 5)(592 + 5).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24414
244142± 5 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24441
244412 = 597362481 is a square with different digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24480
244802 = (22 - 1)(32 - 1)(92 - 1)(162 - 1)(352 - 1) = (22 - 1)(42 - 1)(72 - 1)(162 - 1)(332 - 1)
= (42 - 1)(112 - 1)(5772 - 1) = (52 - 1)(92 - 1)(162 - 1)(352 - 1) = (72 - 1)(352 - 1)(1012 - 1).
by Yoshio Mimura, Kobe, Japan
24485
244852 = 12972 + 12992 + 13012 + 13032 + ... + 17932.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24508
245082 = 600642064 is a square with even digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24514
245142 = 2905 x 2906 + 2907 x 2908 + 2909 x 2910 + 2911 x 2912 + ... + 3039 x 3040.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24576
1602 + 24576 = 2242, 1602 - 24576 = 322.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24598
245982± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24603
246032± 2 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24621
246212 = (162 + 5)(15242 + 5).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24622
246222 = 606242884 is a square with even digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24624
246242 = (12 + 8)(102 + 8)(122 + 8)(642 + 8) = (12 + 8)(22 + 8)(42 + 8)(72 + 8)(642 + 8)
= (12 + 8)(42 + 8)(262 + 8)(642 + 8) = (12 + 8)(42 + 8)(72 + 8)(82 + 8)(262 + 8)
= (12 + 8)(72 + 8)(82 + 8)(102 + 8)(122 + 8) = (42 + 8)(72 + 8)(102 + 8)(642 + 8).
by Yoshio Mimura, Kobe, Japan
24647
24647, 24648, 24649, 24650, 24651 and 24652 are six consecutive integers having square factors (the second case).
Page of Squares : First Upload December 14, 2013 ; Last Revised December 14, 2013by Yoshio Mimura, Kobe, Japan
24648
246482 = (12 + 3)(32 + 3)(322 + 3)(1112 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24650
246502 = (12 + 1)(22 + 1)(122 + 1)(172 + 1)(382 + 1) = (12 + 1)(22 + 1)(42 + 1)(122 + 1)(1572 + 1)
= (122 + 1)(132 + 1)(1572 + 1) = (22 + 1)(42 + 1)(122 + 1)(132 + 1)(172 + 1)
= (22 + 1)(42 + 1)(172 + 1)(1572 + 1) = (22 + 1)(72 + 1)(382 + 1)(412 + 1)
= (32 + 1)(122 + 1)(172 + 1)(382 + 1) = (32 + 1)(42 + 1)(122 + 1)(1572 + 1)
= (42 + 4)(52 + 4)(92 + 4)(1112 + 4) = (92 + 4)(242 + 4)(1112 + 4) = (12 + 9)(42 + 9)(15592 + 9)
= (12 + 9)(72 + 9)(222 + 9)(462 + 9) = (42 + 9)(72 + 9)(222 + 9)(292 + 9).
by Yoshio Mimura, Kobe, Japan
24651
246512 = (32 + 2)(52 + 2)(92 + 2)(1572 + 2).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24656
246562 = (652 + 7)(3792 + 7).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24663
246632± 2 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24676
246762± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24723
247232 = (12 + 8)(192 + 8)(4292 + 8).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24737
247372 = 611919169 is a square which consists of 3 kinds of digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24738
247382 = (22 + 3)(42 + 3)(422 + 3)(512 + 3) = (42 + 3)(112 + 3)(122 + 3)(422 + 3)
= (42 + 3)(422 + 3)(1352 + 3).
by Yoshio Mimura, Kobe, Japan
24752
247522± 3 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24765
247652 = 11752 + 11762 + 11772 + ... + 15122.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24771
247712± 2 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24786
247862 = (12 + 2)(22 + 2)(222 + 2)(2652 + 2) = (12 + 2)(22 + 2)(42 + 2)(52 + 2)(2652 + 2)
= (42 + 2)(222 + 2)(2652 + 2).
by Yoshio Mimura, Kobe, Japan
24807
248072 = 615387249 is a square with different digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24808
(248082 + 2) = (32 + 2)(52 + 2)(92 + 2)(122 + 2)(132 + 2).
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
248220
248222± 5 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24843
248432 = (22 + 3)(62 + 3)(122 + 3)(1242 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24846
248462 = 617323716 is a palindromic square.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24852
248522 = (152 + 3)(182 + 3)(912 + 3) = (32 + 3)(42 + 3)(182 + 3)(912 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24865
The quadratic polynomial 24865X2 - 131210X + 169849 takes the values 2522, 832, 22, 2072, 3682, 5272 at X = 1, 2,..., 6.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24873
248732± 2 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24878
248782 = (22 + 3)(94032 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24908
249082 = 620408464 is a square with even digits.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24912
249122± 5 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24924
249242 = (32 + 3)(82 + 3)(8792 + 3).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24926
249262 = 6152 + 6172 + 6192 + 6212 + ... + 15812.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24942
249422± 5 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24960
249602 = (32 - 1)(52 - 1)(142 - 1)(1292 - 1) = (32 - 1)(72 - 1)(252 - 1)(512 - 1).
249602= 569 x 570 + 570 x 571 + 571 x 572 + 572 x 573 + ... + 1270 x 1271.
1782 + 24960 = 2382, 1782 - 24960 = 822.
Page of Squares : First Upload March 31, 2012 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24961
249612 = 604 + 654 + 1564, the second primitive example.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24966
249662 = (12 + 2)(22 + 2)(132 + 2)(4502 + 2) = (12 + 2)(322 + 2)(4502 + 2)
= (12 + 2)(52 + 2)(62 + 2)(4502 + 2) = (122 + 2)(132 + 2)(1582 + 2)
= (42 + 2)(132 + 2)(4502 + 2).
by Yoshio Mimura, Kobe, Japan
24975
249752 = (62 + 9)(182 + 9)(2042 + 9).
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24981
249812± 2 are primes.
Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24982
249822 = 624100324, and 624100 = 7902, 324 = 182.
Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012by Yoshio Mimura, Kobe, Japan
24990
249902 = (13 + 6)(43 + 6)(93 + 6)(123 + 6).
249902 = 1172 x 1173 + 1174 x 1175 + 1176 x 1177 + 1178 x 1179 + ... + 1748 x 1749.
249902 = (12 + 23 + 33 + ... + 242)(252 + 263 + 273 + ... + 732).
Page of Squares : First Upload March 31, 2012 ; Last Revised February 12, 2014by Yoshio Mimura, Kobe, Japan
24992
249922 = 624600064 is a square with even digits.
249922 = (12 + 7)(22 + 7)(32 + 7)(82 + 7)(792 + 7) = (12 + 7)(32 + 7)(22092 + 7)
= (12 + 7)(82 + 7)(132 + 7)(792 + 7) = (112 + 7)(22092 + 7) = (22 + 7)(82 + 7)(112 + 7)(792 + 7)
= (32 + 7)(82 + 7)(92 + 7)(792 + 7).
by Yoshio Mimura, Kobe, Japan