4400
A cubic polynomial :
(X + 16202)(X + 20792)(X + 44002) = X3 + 51292X2 + 120760202X + 148191120002.
by Yoshio Mimura, Kobe, Japan
4401
44012 = 273 + 1713 + 2433.
44012 = 19368801, 12 + 92 + 32 + 62 + 82 + 82 + 02 + 12 = 162.
44015 = 1651037140033622001 : 12 + 62 + 52 + 12 + 02 + 32 + 72 + 12 + 42 + 02 + 02 + 32 + 32 + 622 + 202 + 02 + 12 = 4401.
Page of Squares : First Upload September 10, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4403
44032 = S2(213) + S2(364), where S2(n) 12 + 22 + 32 + 42 + ... n2.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4404
44042 = 463 + 1643 + 2463.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4407
44072 = 423 + 1213 + 2603 = 493 + 1203 + 2603.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4409
44092 = 19439281, 1 + 9 + 4392 + 8 - 1 = 1 * 9 + 4392 + 8 * 1 = 4409.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4410
44102 = (12 + 5)(52 + 5)(102 + 5)(322 + 5) = (102 + 5)(172 + 5)(252 + 5)
= (32 + 5)(102 + 5)(1152 + 5) = (32 + 5)(42 + 5)(102 + 5)(252 + 5)
= (42 + 5)(52 + 5)(102 + 5)(172 + 5) = (52 + 5)(252 + 5)(322 + 5).
44102 = 73 + 1683 + 2453.
Page of Squares : First Upload August 29, 2008 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4416
44162 = 19501056, 12 + 92 + 52 + 02 + 12 + 02 + 52 + 62 = 132.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4418
44182 = 474 + 474 + 474 + 474.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4420
44202 = (12 + 1)(22 + 1)(32 + 1)(42 + 1)(52 + 1)(212 + 1)
= (12 + 1)(22 + 1)(52 + 1)(132 + 1)(212 + 1) = (12 + 1)(32 + 1)(212 + 1)(472 + 1)
= (12 + 1)(32 + 1)(42 + 1)(52 + 1)(472 + 1) = (12 + 1)(42 + 1)(52 + 1)(72 + 1)(212 + 1)
= (12 + 1)(52 + 1)(132 + 1)(472 + 1) = (32 + 1)(52 + 1)(132 + 1)(212 + 1)
= (32 + 4)(162 + 4)(762 + 4) = (32 + 4)(82 + 4)(92 + 4)(162 + 4) = (82 + 4)(5362 + 4).
Komachi equation: 44202 = 12 / 22 * 342 * 52 * 62 * 782 / 92.
4420k + 7072k + 11713k + 25636k are squares for k = 1,2,3 (2212, 293932, 43468492).
Page of Squares : First Upload October 13, 2010 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4421
44212 = 19545241, 12 + 92 + 52 + 42 + 52 + 22 + 42 + 12 = 132.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4423
44232 = 19562929, a zigzag square.
44232 = 19562929, and 192 + 562 + 22 + 92 + 292 = 192 + 562 + 292 + 22 + 92.
Page of Squares : First Upload September 10, 2007 ; Last Revised November 25, 2008by Yoshio Mimura, Kobe, Japan
4424
44242 = 19571776, a square with odd digits except the last digit 6.
Page of Squares : First Upload August 24, 2013 ; Last Revised August 24, 2013by Yoshio Mimura, Kobe, Japan
4425
44252 = 1103 + 1903 + 2253 = 354 + 404 + 404 + 604.
44252 = 19580625, a zigzag square.
1 / N = 0.000225..., and 225 = 152, where N = 4425, 4426,..., 4444.
Page of Squares : First Upload September 10, 2007 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4428
4428 = (12 + 22 + 32 + ... + 402) / (12 + 22).
4428 = (12 + 8)(102 + 8)(1422 + 8) = (12 + 8)(22 + 8)(192 + 8)(222 + 8)
= (102 + 8)(192 + 8)(222 + 8).
Komachi equations:
44282 = 1232 * 42 * 562 / 72 / 82 * 92 = 1232 * 42 / 562 * 72 * 82 * 92.
by Yoshio Mimura, Kobe, Japan
4431
44315 = 1708082134123691151 : 172 + 02 + 82 + 02 + 82 + 22 + 12 + 342 + 12 + 22 + 32 + 62 + 92 + 112 + 512 = 4431.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4432
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4432 - 2960 - 4441 - 3617 - ... - 4163 - 5650 - 5636 - 4432
(Note f(4432) = 442 + 322 = 2960, f(2960) = 292 + 602 = 4441, etc. See 41)
44322 = (70 + 71 + 72 + 73 + 74 + 75 + 76 + 77)2 + (78 + 79 + 80 + 81 + 82 + 83 + 84 + 85)2 + (86 + 87 + 88 + 89 + 90 + 91 + 92 + 93)2 + ... + (190 + 191 + 192 + 193 + 194 + 195 + 196 + 197)2.
Page of Squares : First Upload September 10, 2007 ; Last Revised October 9, 2008by Yoshio Mimura, Kobe, Japan
4433
44332 = 522 + 532 + 542 + 552 + 562 + 572 + ... + 3892.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4437
44372 = 19686969, a zigzag square.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4440
44402 = 413 + 1593 + 2503.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4441
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4441 - 3617 - 1585 - 7450 - ... - 5650 - 5636 - 4432 - 2960 - 4441
(Note f(4441) = 442 + 412 = 3617, f(3617) = 362 + 172 = 1585, etc. See 41)
by Yoshio Mimura, Kobe, Japan
4446
1378k + 3354k + 4446k + 7722k are squares for k = 1,2,3 (1302, 96202, 7672602).
Page of Squares : First Upload June 14, 2011 ; Last Revised June 14, 2011by Yoshio Mimura, Kobe, Japan
4447
44475 = 1739144418275127007 : 172 + 392 + 142 + 442 + 182 + 22 + 72 + 52 + 12 + 22 + 72 + 02 + 02 + 72 = 4447.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4448
44482 = 19784704, a zigzag square.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4449
44492 = 19793601, a zigzag square.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4450
44502 = 54 + 354 + 554 + 554 = 37 + 57 + 57 + 57 + 97 + 97 + 107.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4451
Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4451 - 4537 - 3394 - 9925 - ... - 10474 - 5493 - 11565 - 4451
(Note f(4451) = 442 + 512 = 4537, f(4537) = 452 + 372 = 3394, etc. See 37)
by Yoshio Mimura, Kobe, Japan
4454
44544 = 393550846429456, and 32 + 92 + 352 + 52 + 02 + 82 + 462 + 42 + 292 + 42 + 52 + 62 = 4454.
Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008by Yoshio Mimura, Kobe, Japan
4455
44552 = 19847025, a square with different digits.
44552 = (131 + 132 + 133)2 + (134 + 135 + 136)2 + (137 + 138 + 139)2 + (140 + 141 + 142)2 + ... + (278 + 279 + 280)2.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4456
44562 = 19855936, 1 - 9 + 8 * 5 / 5 * 93 * 6 = 1 - 9 + 8 / 5 * 5 * 93 * 6 = 4456.
Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4456 - 5072 - 7684 - 12832 - ... - 3536 - 2521 - 1066 - 4456
(Note f(4456) = 442 + 562 = 5072, f(5072) = 502 + 722 = 7684, etc. See 41)
by Yoshio Mimura, Kobe, Japan
4457
44572 = 164 + 194 + 564 + 564.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4458
44582 = 293 + 553 + 2703.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4459
44592 = (11 + 12 + 13 + ... + 59)2 + (60 + 61 + 62 + ... + 108)2.
44595 = 1762736355711266299 : 12 + 72 + 62 + 22 + 72 + 32 + 62 + 32 + 52 + 52 + 72 + 112 + 22 + 62 + 622 + 92 + 92 = 12 + 72 + 622 + 72 + 32 + 62 + 32 + 52 + 52 + 72 + 112 + 22 + 62 + 62 + 22 + 92 + 92 = 4459.
Page of Squares : First Upload September 10, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4460
44602 = 417 x 418 + 419 x 420 + 421 x 422 + 423 x 424 + 425 x 426 + ... + 575 x 576.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4464
44642 = 623 + 1363 + 2583 = 1033 + 1533 + 2483.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4465
44652 + 44662 + 44672 + ... + 45122 = 45132 + 45142 + 45152 + ... + 45592.
Page of Squares : First Upload September 13, 2011 ; Last Revised September 13, 2011by Yoshio Mimura, Kobe, Japan
4466
44662 = 19945156, and
1 + 99 * 45 - 1 + 5 + 6 = 1 * 99 * 45 + 1 * 5 + 6 = 1 * 99 * 45 * 1 + 5 + 6 = 4466.
by Yoshio Mimura, Kobe, Japan
4467
44672 = 333 + 1553 + 2533.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan
4469
44695 = 1782591261823093349 : 12 + 72 + 82 + 252 + 92 + 122 + 62 + 12 + 82 + 22 + 302 + 92 + 32 + 32 + 492 = 4469.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4472
44722 = 693 + 703 + 713 + 723 + 733 + 743 + ... + 1003.
Page of Squares : First Upload November 25, 2008 ; Last Revised November 25, 2008by Yoshio Mimura, Kobe, Japan
4480
A cubic polynomial :
(X + 30962)(X + 32252)(X + 44802) = X3 + 63292X2 + 223789202X + 447310080002.
Komachi equation: 44802 = 12 * 22 * 32 * 42 * 52 * 62 * 72 * 82 / 92.
Page of Squares : First Upload September 10, 2007 ; Last Revised October 13, 2010by Yoshio Mimura, Kobe, Japan
4485
44852 = 20115225, 22 + 02 + 12 + 12 + 52 + 22 + 22 + 52 = 82.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4486
44865 = 1816754925203822176 : 182 + 162 + 72 + 52 + 492 + 252 + 202 + 32 + 82 + 22 + 22 + 172 + 62 = 4486.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4488
44882 = (52 + 8)(162 + 8)(482 + 8).
44882 = 333 + 953 + 2683.
Page of Squares : First Upload August 29, 2008 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4489
The square of 67.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4494
The square root of 4494 is 67.0373...., and 67 = 02 + 32 + 72 + 32.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4495
44952 = 20205025, a square with just 3 kinds of digits.
Page of Squares : First Upload September 10, 2007 ; Last Revised September 10, 2007by Yoshio Mimura, Kobe, Japan
4497
44972 = 324 + 344 + 384 + 634.
Page of Squares : First Upload August 29, 2008 ; Last Revised August 29, 2008by Yoshio Mimura, Kobe, Japan