Squares:2600s

2600

2600² = (1² + 4)(10² + 4)(114² + 4) = (1² + 4)(2² + 4)(14² + 4)(29² + 4)
= (1² + 4)(2² + 4)(3² + 4)(10² + 4)(11² + 4) = (1² + 4)(2² + 4)(3² + 4)(114² + 4)
= (3² + 4)(6² + 4)(10² + 4)(11² + 4) = (3² + 4)(6² + 4)(114² + 4)
= (4² + 4)(16² + 4)(36² + 4) = (6² + 4)(14² + 4)(29² + 4) = (2⁴ + 4)(4⁴ + 4)(6⁴ + 4).

Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2601

The square of 51.

1 / 2601 = 0.0003844..., and 3844 = 62².

408^k + 1156^k + 2601^k + 3060^k are squares for k = 1,2,3 (85², 4199², 218773²).

Page of Squares : First Upload May 1, 2007 ; Last Revised May 20, 2011
by Yoshio Mimura, Kobe, Japan

2604

2604² = (3² + 3)(9² + 3)(82² + 3).

Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2607

2607² = 6796449 appears in the decimal expressions of π:
π = 3.14159•••6796449••• (from the 102403rd digit)

Page of Squares : First Upload November 4, 2008 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2609

Komachi equation: 2609² = 1⁴ / 2⁴ * 34⁴ + 5⁴ - 67⁴ + 8⁴ * 9⁴.

Page of Squares : First Upload September 24, 2010 ; Last Revised September 24, 2010
by Yoshio Mimura, Kobe, Japan

2610

2610² = 564 x 565 + 566 x 567 + 568 x 569 + 570 x 571 + 572 x 573 + ... + 602 x 603.

2610² = (1² + 9)(6² + 9)(123² + 9) = (21² + 9)(123² + 9) = (3² + 9)(4² + 9)(123² + 9)
= (6² + 9)(7² + 9)(51² + 9) = (7² + 9)(9² + 9)(36² + 9).

Page of Squares : First Upload May 1, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2613

The square root of 2613 is 51.117..., and 51 = 1² + 1² + 7².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2616

2616²± 5 are primes.

2616² = 34³ + 129³ + 167³.

Page of Squares : First Upload July 24, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

2617

2617² = 6848689, a square pegged by 8.

2617² = 6848689, 6² + 8² + 4² + 8² + 6² + 8² + 9² = 19².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2618

2618² = 6853924, a square with different digits.

2618² = 23⁴ + 37⁴ + 37⁴ + 41⁴.

2618² = 6853924 appears in the decimal expressions of e:
e = 2.71828•••6853924••• (from the 60086th digit)

Page of Squares : First Upload May 1, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2619

2619² = 6859161, a zigzag square.

2619² = 83² + 84² + 85² + 86² + 87² + 88² + ... + 276².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2621

2621² = 156² + 659² + 2532² = 2352² + 956² + 651².

Page of Squares : First Upload September 7, 2013 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

2624

2624² = 6885376, and 6 * 88 * 5 - 3 - 7 - 6 = 2624.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2625

2625² = 60³ + 70³ + 185³.

Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2626

2626² = (5² + 1)(515² + 1).

Page of Squares : First Upload November 2, 2013 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

2628

2628² = 6906384, a zigzag square.

2628² + 2629² + 2630² + ... + 2664² = 2665² + 2666² + 2667² + ... + 2700².

Komachi equations:
2628² = 9² * 876² * 5² * 4² / 3² / 2² / 10² = 9² * 876² / 5² / 4² / 3² * 2² * 10².

2628² = 57³ + 84³ + 183³ = 24⁴ + 32⁴ + 32⁴ + 46⁴.

Page of Squares : First Upload May 1, 2007 ; Last Revised September 13, 2011
by Yoshio Mimura, Kobe, Japan

2631

2631² = 6922161, and 69 + 2 * 21 * 61 = 2631.

2631² = 6922161 appears in the decimal expressions of π:
π = 3.14159•••6922161••• (from the 66460th digit)

Page of Squares : First Upload May 1, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2632

Komachi equation: 2632² = 987² / 6² / 5² / 4² * 32² * 10².

Page of Squares : First Upload September 24, 2010 ; Last Revised September 24, 2010
by Yoshio Mimura, Kobe, Japan

2634

2634²± 5 are primes.

2634² = 6937956 appears in the decimal expressions of π:
π = 3.14159•••6937956••• (from the 26353rd digit)
(6937956 is the eighth 7-digit square in the expression of π.)

Page of Squares : First Upload November 4, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

2635

The square root of 2635 is 51.33225..., and 51 = 3² + 3² + 2² + 2² + 5².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2636

2636² = 6948496, a palindromic and zigzag square.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2639

A cubic polynomial :
(X + 1044²)(X + 2639²)(X + 4368²) = X³ + 5209²X² + 12698868²X + 12034346688².

2639² = 6964321 appears in the decimal expressions of π:
π = 3.14159•••6964321••• (from the 73594th digit)

Page of Squares : First Upload May 1, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2640

2640² = 1³ + 12³ + 191³ = 2³ + 48³ + 190³ = 9³ + 10³ + 191³ = 34³ + 144³ + 158³ = 60³ + 148³ + 152³.

2640² = (11² - 1)(241² - 1) = (3² - 1)(4² - 1)(241² - 1).

2640² = (1²)(2² + 3² + 4² + 5² + 6²)(7² + 8² + 9² + ... + 61²).

Page of Squares : First Upload July 24, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2644

264452644 = 16262².

2644² = 3⁰ + 3¹ + 3³ + 3⁴ + 3⁵ + 3⁶ + 3⁷ + 3⁹ + 3¹⁰ + 3¹² + 3¹³ + 3¹⁴.

Page of Squares : First Upload May 1, 2007 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

2645

2645² = 1³ + 120³ + 174³ = 25³ + 134³ + 166³.

2645² = (58 + 59 + 60 + ... + 80)² + (81 + 82 + 83 + ... + 103)² + (104 + 105 + 106 + ... + 126)² + ... + (81 + 82 + 83 + ... + 103)².

Page of Squares : First Upload May 1, 2007 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2646

2646² = 63³ + 189³ = 9³ + 126³ + 171³.

The integral triangle of sides 449, 117657, 118090 has square area 2646².

2646² = (1² + 5)(2² + 5)(11² + 5)(32² + 5) = (1² + 5)(4² + 5)(7² + 5)(32² + 5)
= (2² + 5)(3² + 5)(7² + 5)(32² + 5) = (2² + 5)(4² + 5)(11² + 5)(17² + 5) = (7² + 5)(11² + 5)(32² + 5).

Komachi equations:
2646² = 1² / 2² * 3² * 4² * 56² * 7² / 8² * 9² = 9² * 8² * 7² / 6² / 5² / 4² * 3² * 210²
= 9² / 8² * 7² * 6² / 5² * 4² / 3² * 210² = 98² / 7² * 6² / 5² / 4² * 3² * 210².

Page of Squares : First Upload July 24, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2647

2647² = 67³ + 127³ + 167³.

Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2648

2648² = 8⁵ + 18⁵ + 18⁵ + 20⁵.

Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2650

2650² = (2² + 6)(838² + 6) = (7² + 4)(364² + 4).

2650² = (43 + 44 + 45 + 46 + 47)² + (48 + 49 + 50 + 51 + 52)² + (53 + 54 + 55 + 56 + 57)² + ... + (158 + 159 + 160 + 161 + 162)².

Page of Squares : First Upload May 1, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2652

2652² = 94³ + 134³ + 156³.

2550^k + 2652^k + 6681^k + 11526^k are squares for k = 1,2,3 (153², 13821², 1365525²).

Page of Squares : First Upload July 24, 2008 ; Last Revised May 20, 2011
by Yoshio Mimura, Kobe, Japan

2654

2654² = 7043716, a zigzag square.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2656

2656² = 7054336, 7² + 0² + 5² + 4² + 3² + 3² + 6² = 12².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2657

2657² = 16⁴ + 28⁴ + 28⁴ + 49⁴.

1 / 2657 = 0.00037636... and 37636 = 194².

Page of Squares : First Upload May 1, 2007 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2660

2660²± 3 are primes.

2660² = 21³ + 110³ + 179³.

Page of Squares : First Upload July 24, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

2661

2661² = 86³ + 142³ + 153³.

Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2662

2662² = 11⁶ + 11⁶ + 11⁶ + 11⁶.

2662² = 3⁰ + 3¹ + 3² + 3³ + 3⁴ + 3⁵ + 3¹¹ + 3¹² + 3¹³ + 3¹⁴.

Page of Squares : First Upload July 24, 2008 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

2669

2669² = S₂(11) + S₂(277), where S₂(n) = 1² + 2² + 3² + ... + n².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2672

2672² = 7139584, a square with different digits.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2678

2678² = 7171684 appears in the decimal expressions of e:
e = 2.71828•••7171684••• (from the 32712nd digit)
(7171684 is the eighth 7-digit square in the expression of e.)

Page of Squares : First Upload May 1, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2679

A cubic polynomial :
(X + 448²)(X + 684²)(X + 2679²) = X² + 32801²X² + 2211828²X + 820931328².

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2680

Komachi equations:
2680² = 12² * 3² / 4² + 5² * 67² * 8² - 9² = 12² * 3² / 4² * 5² * 67² * 8² / 9²
= - 12² * 3² / 4² + 5² * 67² * 8² + 9².

Page of Squares : First Upload July 24, 2008 ; Last Revised September 24, 2010
by Yoshio Mimura, Kobe, Japan

2682

2682² = 1⁴ + 1⁴ + 39⁴ + 47⁴ = 5⁷ + 7⁷ + 8⁷ + 8⁷ + 8⁷.

Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2684

2684² = 7203856, a square with different digits.

1 / 2684 = 0.000372578241..., and 37² + 25² + 7² + 8² + 24² + 1² = 2684.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2686

2686² = 7214596, a square with different digits.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2687

2687² = 23³ + 77³ + 189³.

1 / 2687 = 0.0003721... and 3721 = 61².

Page of Squares : First Upload May 1, 2007 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2688

2688² = (3² - 1)(5² - 1)(13² - 1)(15² - 1).

Cubic polynomials ;
(X + 828²)(X + 2688²)(X + 4531²) = X³ + 5333²X² + 12936948²X + 10084483584²,
(X + 1504²)(X + 2688²)(X + 3807²) = X³ + 4897²X² + 12403488²X + 15390756864²,
(X + 1935²)(X + 2688²)(X + 2800²) = X³ + 4337²X² + 10632720²X + 14563584000².

Page of Squares : First Upload May 1, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2689

2689 is the 6th prime for which the Legendre symbol (a / 2689) = 1 for a = 1,2,...,12.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2691

2691² = 81³ + 88³ + 182³.

Page of Squares : First Upload July 24, 2008 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2694

2694² = 1! + 1! + 2! + 2! + 3! + 4! + 10! + 10! = 2! + 2! + 2! + 3! + 4! + 10! + 10!
= 3! + 3! + 3! + 3! + 3! + 3! + 10! + 10! = 3! + 3! + 4! + 10! + 10!.

2694² = 7257636, and 7 * 2 * 576 / 3 + 6 = 2694.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan

2695

2695² = 1! + 4! + 5! + 5! + 5! + 7! + 10! + 10!.

2695^k + 10615^k + 12485^k + 22605^k are squares for k = 1,2,3 (220², 28050², 3835700²).

Page of Squares : First Upload May 1, 2007 ; Last Revised May 20, 2011
by Yoshio Mimura, Kobe, Japan

2696

2696² = 38³ + 92³ + 186³.

2696² = 7268416, and 7 + 2684 - 1 + 6 = 2696.

Page of Squares : First Upload May 1, 2007 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2697

2697² = 7273809, a zigzag square.

2697² = 2⁴ + 34⁴ + 41⁴ + 42⁴.

2697² = 7273809, 7² + 2² + 7² + 3² + 8² + 0² + 9² = 16².

Page of Squares : First Upload May 1, 2007 ; Last Revised July 24, 2008
by Yoshio Mimura, Kobe, Japan

2699

2699² = 7284601, a zigzag square with different digits.

Page of Squares : First Upload May 1, 2007 ; Last Revised May 1, 2007
by Yoshio Mimura, Kobe, Japan