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24000 - 24999

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, 2012
by 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, 2012
by Yoshio Mimura, Kobe, Japan

24010

240102± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24018

240182± 5 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24020

240202 = 576960400, and 576 = 242, 960400 = 9802.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by 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, 2014
by Yoshio Mimura, Kobe, Japan

24035

240352 = 577681225, and 5776 = 762, 81225 = 2852.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by 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, 2012
by 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).

Page of Squares : First Upload November 9, 2013 ; Last Revised February 12, 2014
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, 2014
by Yoshio Mimura, Kobe, Japan

24154

241542± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24188

241882± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
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, 2014
by Yoshio Mimura, Kobe, Japan

24210

242102 = (32 + 9)(232 + 9)(2462 + 9).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24242

242422± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2012
by Yoshio Mimura, Kobe, Japan

24288

242882 = (232 - 1)(10572 - 1).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2012
by Yoshio Mimura, Kobe, Japan

24350

243502 = 592922500, and 5929 = 772, 22500 = 1502.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by 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).

Page of Squares : First Upload August 31, 2013 ; Last Revised February 12, 2014
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, 2014
by 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, 2014
by 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, 2014
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
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, 2014
by Yoshio Mimura, Kobe, Japan

24414

244142± 5 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2012
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24485

244852 = 12972 + 12992 + 13012 + 13032 + ... + 17932.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by 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, 2012
by 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, 2012
by Yoshio Mimura, Kobe, Japan

24576

1602 + 24576 = 2242, 1602 - 24576 = 322.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by Yoshio Mimura, Kobe, Japan

24598

245982± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24603

246032± 2 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24621

246212 = (162 + 5)(15242 + 5).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2012
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
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, 2013
by 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, 2014
by 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).

Page of Squares : First Upload November 9, 2013 ; Last Revised February 12, 2014
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, 2014
by Yoshio Mimura, Kobe, Japan

24656

246562 = (652 + 7)(3792 + 7).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24663

246632± 2 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24676

246762± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24723

247232 = (12 + 8)(192 + 8)(4292 + 8).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2012
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24752

247522± 3 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24765

247652 = 11752 + 11762 + 11772 + ... + 15122.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by Yoshio Mimura, Kobe, Japan

24771

247712± 2 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
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, 2012
by 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, 2012
by Yoshio Mimura, Kobe, Japan

248220

248222± 5 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2014
by Yoshio Mimura, Kobe, Japan

24846

248462 = 617323716 is a palindromic square.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by 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, 2014
by 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, 2012
by Yoshio Mimura, Kobe, Japan

24873

248732± 2 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24878

248782 = (22 + 3)(94032 + 3).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2012
by Yoshio Mimura, Kobe, Japan

24912

249122± 5 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24924

249242 = (32 + 3)(82 + 3)(8792 + 3).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24926

249262 = 6152 + 6172 + 6192 + 6212 + ... + 15812.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by Yoshio Mimura, Kobe, Japan

24942

249422± 5 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by 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, 2014
by 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, 2012
by 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).

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
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, 2014
by Yoshio Mimura, Kobe, Japan

24981

249812± 2 are primes.

Page of Squares : First Upload February 12, 2014 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan

24982

249822 = 624100324, and 624100 = 7902, 324 = 182.

Page of Squares : First Upload March 31, 2012 ; Last Revised March 31, 2012
by 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, 2014
by 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).

Page of Squares : First Upload March 31, 2012 ; Last Revised February 12, 2014
by Yoshio Mimura, Kobe, Japan