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4900 - 4999

4900

The square of 70.

49002 = 352 + 2102 + 2452 = 812 + 1672 + 2662

Komachi equations:
49002 = 14 * 24 * 34 * 44 * 54 * 64 * 74 / 84 / 94 = 14 * 24 / 34 / 44 * 54 / 64 * 74 * 84 * 94
 = 14 / 24 * 34 / 44 * 54 * 64 * 74 * 84 / 94 = 14 / 24 * 34 * 454 / 64 * 74 * 84 / 94
 = 94 * 84 * 74 / 64 * 54 / 44 / 34 * 24 */ 14 = 984 / 74 * 64 * 54 / 44 / 34 * 24 */ 14
 = 984 / 74 / 64 * 54 * 44 * 34 / 24 */ 14.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2010
by Yoshio Mimura, Kobe, Japan

4901

49012=24019801, and 2401 = 492, 9801 = 992.

The square root of 4901 is 70.007142..., and 70 = 02 + 02 + 72 + 12 + 42 + 22.

49012 = (254 + 255 + 256 + ... + 266)2 + (267 + 268 + 269 + ... + 279)2.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4902

49022 = 822 + 2092 + 2432

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

4903

1 / 4903 = 0.00020395676116, 22 + 02 + 392 + 562 + 72 + 62 + 112 + 62 = 4903,
1 / 4903 = 0.00020395676116, 22 + 0392 + 562 + 72 + 62 + 112 + 62 = 4903.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4904

49042 = 42 + 1282 + 2802

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

4906

49062 = 24068836, and 2 + 4068 + 836 = 4906.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4907

49075 = 2844987023923141307 : 22 + 82 + 42 + 492 + 82 + 72 + 02 + 22 + 392 + 232 + 142 + 12 + 32 + 02 + 72 = 4907.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4908

49082± 5 are primes.

49082 = 182 + 1522 + 2742

49082 = 24088464, a square with even digits.

Page of Squares : First Upload October 15, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4913

49132 = 572 + 1262 + 2802 = 42 + 262 + 562 + 612 = 82 + 222 + 442 + 672 = 82 + 462 + 472 + 622 = 172 + 342 + 342 + 682

49132 = 24137569, a square with different digits.

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4913 - 2570 - 5525 - 3650 - ... - 41 - 1681 - 6817 - 4913
(Note f(4913) = 492 + 132 = 2570,   f(2570) = 252 + 702 = 5525, etc. See 41)

A cubic polynomial :
(X + 19202)(X + 27372)(X + 36002) = X3 + 49132X2 + 131330402X + 189181440002.

49132 is the first sum of four 4th powers in four ways:
  49132 = 44 + 264 + 564 + 614 = 84 + 224 + 444 + 674 = 84 + 464 + 474 + 624
  = 174 + 344 + 344 + 684.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

4914

49142 = 813 + 823 + 833 + 843 + 853 + 863 + ... + 1083.

Komachi equation: 49142 = 92 * 82 * 72 * 652 / 42 * 32 * 22 / 102.

Page of Squares : First Upload November 25, 2008 ; Last Revised October 15, 2010
by Yoshio Mimura, Kobe, Japan

4915

49152 = 182 + 242 + 2892

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

4916

49162 = 32 + 82 + 132 + 262 + 262

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

4917

4917 = (12 + 22 + 32 + ... + 2972) / (12 + 22 + 32 + ... + 172).

Page of Squares : First Upload November 25, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

4918

49182 = 24186724, a zigzag square.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4920

49202 = 162 + 682 + 2882 = 1062 + 1982 + 2482

The square root of 4920 is 70.1427..., and 70 = 12 + 42 + 22 + 72.

Page of Squares : First Upload October 15, 2007 ; Last Revised September 4, 2008
by Yoshio Mimura, Kobe, Japan

4922

49222 = 12 + 1162 + 2832

49222 = 24226084, and 22 + 42 + 22 + 22 + 62 + 02 + 82 + 42 = 122.

49222 = 24226084, a square consisting of even digits.

Page of Squares : First Upload October 15, 2007 ; Last Revised September 4, 2008
by Yoshio Mimura, Kobe, Japan

4924

49242± 3 are primes.

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

4928

49282± 3 are primes.

49282 = 24285184, 24 / 2 * 8 * 51 + 8 * 4 = 4928.

49282 = (22 + 7)(52 + 7)(72 + 7)(352 + 7) = (22 + 7)(72 + 7)(92 + 7)(212 + 7)
= (72 + 7)(212 + 7)(312 + 7).

Page of Squares : First Upload October 15, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4929

49292 = 22 + 542 + 2892

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

4930

4930 = (12 + 22 + 32 + ... + 1442) / (12 + 22 + 32 + ... + 82).

49302 = (12 + 1)(22 + 1)(382 + 1)(412 + 1) = (22 + 1)(42 + 1)(132 + 1)(412 + 1)
= (32 + 1)(382 + 1)(412 + 1) = (52 + 4)(82 + 4)(1112 + 4)
= (12 + 9)(15592 + 9) = (42 + 9)(52 + 9)(72 + 9)(222 + 9) = (72 + 9)(222 + 9)(292 + 9).

4930k + 11078k + 15254k + 22562k are squares for k = 1,2,3 (2322, 298122, 40637122).

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4930 - 3301 - 1090 - 8200 - ... - 10600 - 37 - 1369 - 4930
(Note f(4930)= 492 + 302 = 3301,   f(3301) = 332 + 012 = 1090, etc. See 37)

Page of Squares : First Upload October 9, 2008 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4931

49312 = 24314761 appears in the decimal expressions of e:
  e = 2.71828•••24314761••• (from the 41740th digit)
  (24314761 is the tenth 8-digit square in the expression of e.)

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

4932

49322 = 1862 + 1962 + 2182 = 122 + 122 + 482 + 662

49325 = 2918201925080466432 : 292 + 182 + 202 + 12 + 92 + 22 + 52 + 02 + 82 + 02 + 462 + 62 + 42 + 322 = 4932.

Page of Squares : First Upload September 4, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4935

49352 = 212 + 1792 + 2652

Komachi equations:
49352 = 9872 * 62 * 52 / 42 / 32 * 22 */ 12 = 9872 / 62 * 52 * 42 * 32 / 22 */ 12.

Page of Squares : First Upload September 4, 2008 ; Last Revised October 15, 2010
by Yoshio Mimura, Kobe, Japan

4936

1 / 4936 = 0.0002025, 2025 = 452. (also 4937 and 4938)

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4937

49372 = 24373969, a zigzag square.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4940

2812k + 4940k + 8056k + 13433k are squares for k = 1,2,3 (1712, 166632, 17577092).

Page of Squares : First Upload June 17, 2011 ; Last Revised June 17, 2011
by Yoshio Mimura, Kobe, Japan

4941

49412 = 1112 + 2112 + 2392 = 1842 + 1852 + 2282

49412 = 13 - 23 + 33 - 43 + 53 - 63 + ... +3633 - 3643 + 3653.

Page of Squares : First Upload October 15, 2007 ; Last Revised September 4, 2008
by Yoshio Mimura, Kobe, Japan

4944

49442 = 1322 + 2082 + 2362

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

4946

49462 = 24462916, and 2446 * 2 + 9 * 1 * 6 = 4946.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4947

1 / 4947 = 0.00020214271275520, 202 + 22 + 142 + 272 + 122 + 72 + 552 + 202 = 4947.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4948

49482 = 24482704, and 22 + 42 + 42 + 82 + 22 + 72 + 02 + 42 = 132.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4949

49492 = 24492601, and 24 + 4926 + 0 - 1 = 4949.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4950

49502 = 1292 + 2192 + 2282

The square root of 4950 is 70.356..., and 70 = 32 + 52 + 62.

49502 = 24502500, 2450 * 2 + 50 + 0 = 2450 + 2500 = 4950.

Page of Squares : First Upload October 15, 2007 ; Last Revised September 4, 2008
by Yoshio Mimura, Kobe, Japan

4955

49554 = 602801931600625, and 602 + 22 + 82 + 02 + 192 + 32 + 162 + 02 + 02 + 62 + 252 = 4955.

Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

4957

49572 = 24571849, and 22 + 42 + 52 + 72 + 12 + 82 + 42 + 92 = 162.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4966

49662 = 42 + 62 + 142 + 252 + 272

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

4967

49672 = 82 + 142 + 462 + 672

49672 = 24671089, a square with different digits.

Page of Squares : First Upload October 15, 2007 ; Last Revised September 4, 2008
by Yoshio Mimura, Kobe, Japan

4971

49712 = 1802 + 1842 + 2332

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

4972

49722 = 52 + 62 + 112 + 122 + 302

1 / 4972 = 0.000201126307320, 202 + 12 + 122 + 632 + 02 + 72 + 32 + 202 = 4972,
1 / 4972 = 0.000201126307320, 202 + 12 + 122 + 632 + 072 + 32 + 202 = 4972.

Page of Squares : First Upload October 15, 2007 ; Last Revised September 4, 2008
by Yoshio Mimura, Kobe, Japan

4973

Loop of length 6 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4973 -- 7730 -- 6829 -- 5465 -- 7141 -- 6722 -- 4973
(Note f(4973) = 492 + 732 = 7730,   f(7730) = 772 + 302 = 6829, etc. See 41)

Page of Squares : First Upload October 9, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

4976

49762= 112 x 113 + 113 x 114 + 114 x 115 + 115 x 116 + ... + 422 x 423.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4977

49772 = 24770529, and 24 + 7 * 705 + 2 * 9 = 4977.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4978

49782 = 192 + 862 + 2892 = 812 + 1712 + 2682

2242k + 4978k + 13262k + 15618k are squares for k = 1,2,3 (1902, 212042, 25053402).

Page of Squares : First Upload September 4, 2008 ; Last Revised June 17, 2011
by Yoshio Mimura, Kobe, Japan

4980

The quadratic polynomial -4980X2 + 42060X - 34271 takes the values 532, 1732, 2172, 2332, 2272, 1972, 1272 at X = 1, 2,..., 7,

Page of Squares : First Upload December 15, 2008 ; Last Revised December 15, 2008
by Yoshio Mimura, Kobe, Japan

4983

49832 = 662 + 742 + 2892 = 1132 + 1982 + 2502

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

4985

49852 = 252 + 1862 + 2642 = 812 + 1872 + 2612

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

4986

49862± 5 are primes.

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

4987

49872 = 24870169, a square with different digits.

A cubic polynomial :
(X + 15042)(X + 26882)(X + 38072) = X3 + 48972X2 + 124034882X + 153907568642.

Page of Squares : First Upload October 15, 2007 ; Last Revised October 15, 2007
by Yoshio Mimura, Kobe, Japan

4988

1505k + 4042k + 4988k + 6106k are squares for k = 1,2,3 (1292, 89872, 6489992).

1 / 4988 = 0.0002004811547714, 22 + 02 + 0482 + 12 + 152 + 472 + 72 + 142 = 4988,
1 / 4988 = 0.0002004811547714, 22 + 002 + 482 + 12 + 152 + 472 + 72 + 142 = 4988,
1 / 4988 = 0.0002004811547714, 22 + 00482 + 12 + 152 + 472 + 72 + 142 = 4988.

Page of Squares : First Upload October 15, 2007 ; Last Revised June 17, 2011
by Yoshio Mimura, Kobe, Japan

4992

49922 = 1002 + 1562 + 2722 = 1282 + 2082 + 2402

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

4998

Komachi equation: 49982 = 982 * 7652 * 42 / 32 / 22 / 102.

4998k + 6468k + 10542k + 13713k are squares for k = 1,2,3 (1892, 191312, 20360972).
4998k + 6888k + 8106k + 15729k are squares for k = 1,2,3 (1892, 196352, 22080872).

Page of Squares : First Upload October 15, 2010 ; Last Revised June 17, 2011
by Yoshio Mimura, Kobe, Japan

4999

49993 = 124925014999, and 12 + 22 + 492 + 22 + 52 + 02 + 12 + 492 + 92 + 92 = 4999.

Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan