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1500 - 1599

1500

15002 = 53 + 903 + 1153.

15002 = (1)(2)(3 + 4 + 5 + ... + 22)(23 + 24 + 25 + ... + 97),
15002 = (1)(2 + 3)(4 + 5 + 6 + ... + 11)(12)(13 + 14 + 15 + ... + 37).
15002 = (1)(2 + 3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 112).
15002 = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 17)(18 + 19 + 20 + ... + 22).

Page of Squares : First Upload February 5, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1501

A cubic polynomial :
(X + 7042)(X + 8612)(X + 10082) = X3 + 15012X2 + 12744482X + 6109931522.

15012 = 2253001,
2 - 2 + 5 * 300 + 1 = 2 / 2 + 5 * 300 * 1 = 2 / 2 * 5 * 300 + 1 = 2 * 25 * 30 + 0 + 1 = 1501.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1502

15022 = 2256004, 2 + 2 * 5 * 600 / 4 = 2 + 25 * 60 + 0 * 4 = 1502.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1503

15032 = 123 + 483 + 1293.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1504

15042 = 563 + 883 + 1123 = 643 + 763 + 1163.

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

15042 = 2262016 appears in the decimal expressions of e:
  e = 2.71828•••2262016••• (from the 72805th digit)

Page of Squares : First Upload February 5, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1505

1505k + 4042k + 4988k + 6106k are squares for k = 1,2,3 (1292, 89872, 6489992).
344k + 1505k + 4558k + 10234k are squares for k = 1,2,3 (1292, 113092, 10816652).

Page of Squares : First Upload April 26, 2011 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1507

15072 = 85 + 95 + 155 + 175.

292358k + 304414k + 739937k + 934340k are squares for k = 1,2,3 (15072, 12643732, 11287113532).

Page of Squares : First Upload July 10, 2008 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1509

75450k + 202206k + 683577k + 1315848k are squares for k = 1,2,3 (15092, 14984372, 16144504292).

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1511

1511 is the 10th prime for which the Legendre symbol (a/p) = 1 for a = 1, 2,..., 10.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1512

15122± 5 are primes.

A+B, A+C, A+D, B+C, B+D and C+D are squares for A =1512, B = 3672, C = 5544, D = 9097.

15122 = 2286144, 228 * 6 + 144 = 1512.

Komachi equation: 15122 = 12 * 2342 * 562 / 782 * 92.

15122 = 35 + 65 + 155 + 155 + 155.

15122 = (1)(2)(3)(4)(5 + 6 + 7 + ... + 436),
15122 = (1)(2)(3)(4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15)(16),
15122 = (1)(2)(3)(4 + 5 + 6 + ... + 24)(25 + 26 + 27 + ... + 56),
15122 = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 571),
15122 = (1)(2)(3 + 4)(5 + 6 + 7)(8)(9)(10 + 11 + 12 + ... + 18),
15122 = (1)(2)(3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14),
15122 = (1)(2)(3 + 4 + 5 + ... + 29)(30 + 31 + 32 + ... + 78),
15122 = (1)(2)(3 + 4 + 5 + 6)(7)(8)(9)(10 + 11 + 12 + ... + 18),
15122 = (1)(2)(3 + 4 + 5 + 6)(7)(8)(9 + 10 + 11 + 12)(13 + 14),
15122 = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 14)(15 + 16 + 17 + ... + 41),
15122 = (1)(2)(3 + 4 + 5 + ... + 9)(10 + 11 + 12 + ... + 233),
15122 = (1)(2 + 3 + 4 + ... + 25)(26 + 27 + 28 + ... + 121),
15122 = (1)(2 + 3 + 4 + 5)(6)(7)(8 + 9 + 10 + ... + 88),
15122 = (1)(2 + 3 + 4 + 5)(6)(7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15),
15122 = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10 + ... + 28)(29 + 30 + 31 + ... + 35),
15122 = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10 + ... + 55)(56),
15122 = (1)(2 + 3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14 + 15),
15122 = (1 + 2)(3)(4 + 5)(6)(7)(8)(9 + 10 + 11 + ... + 15),
15122 = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + ... + 26)(27 + 28 + 29),
15122 = (1 + 2)(3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11 + ... + 15)(16),
15122 = (1 + 2)(3 + 4 + 5 + ... + 9)(10 + 11)(12 + 13 + 14 + 15)(16),
15122 = (1 + 2)(3 + 4 + 5)(6)(7)(8)(9)(10 + 11),
15122 = (1 + 2)(3 + 4 + 5)(6)(7)(8 + 9 + 10 + ... + 55),
15122 = (1 + 2)(3 + 4 + 5)(6)(7 + 8 + 9 + ... + 14)(15 + 16 + 17 + ... + 21),
15122 = (1 + 2 + 3 + ... + 7)(8)(9)(10 + 11)(12 + 13 + 14 + 15),
15122 = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 436),
15122 = (1 + 2 + 3)(4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15)(16),
15122 = (1 + 2 + 3)(4 + 5 + 6 + ... + 24)(25 + 26 + 27 + ... + 56).

Page of Squares : First Upload February 5, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1513

S2(113) + S2(175) = 15132, where S2(n) = 12 + 22 + 32 + ... + n2.

15132 = 2289169, 2 - 2 - 8 + 9 * 169 = 1513.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1515

15152 = 2295225, a square with just 3 kinds of digits 2, 5 and 9.

15152 = 2295225, 2 * 2 * 95 * 2 * 2 - 5 = 1515 .

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1516

82k + 182k + 722k + 1318k are squares for k = 1,2,3 (482, 15162, 516962).

Page of Squares : First Upload April 26, 2011 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1517

1517k + 2257k + 12173k + 33337k are squares for k = 1,2,3 (2222, 355942, 62344262).

Page of Squares : First Upload April 26, 2011 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1518

The integral triangle of sides 5336, 6843, 12155 has square area 15182.

15182 = 2304324, 2304 = 482 and 324 = 182.

15182 = (1 + 2 + 3 + ... + 11)(12 + 13 + 14 + ... + 264).

15182 = (33 + 9)(403 + 9).

15182 = 72 + 82 + 92 + 102 + 112 + 122 + 132 + ... + 1902.

15182 = 2304324 appears in the decimal expressions of π:
  π = 3.14159•••2304324••• (from the 62461st digit)

Page of Squares : First Upload February 5, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1519

15192 = 2307361, a zigzag square.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1520

15202 = 213 + 673 + 1263 = 643 + 743 + 1183.

Komachi equation: 15202 = 92 * 82 * 762 / 542 * 32 / 22 * 102.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1521

The square of 39.

15212 = 2313441, a square every digit of which is non-zero and smaller than 5.

15212 = (356 + 357 + 358)2 + (359 + 360 + 361)2.

15212 = 193 + 893 + 1173.

Page of Squares : First Upload February 5, 2007 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

1522

15222 = 2316484, a zigzag square.

15222± 3 are primes.

Page of Squares : First Upload February 5, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1524

1 / 1524 = 0.0006561... and 6561 = 812.

15242 = 2322576, 2 * 32 * 25 - 76 = 1524.

15242 = 553 + 95 + 87.

Page of Squares : First Upload February 5, 2007 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

1525

15252 = (22 + 1)(6822 + 1).

15252 = 732 + 742 + 752 + 762 + 772 + 782 + 792 + ... +1942.

Page of Squares : First Upload February 5, 2007 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

1526

15262 = 2328676, a zigzag square.

Komachi equations:
15262 = 982 / 72 * 6542 / 32 / 22 */ 12.

Page of Squares : First Upload February 5, 2007 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1527

15272 = 93 + 1003 + 1103 = 163 + 963 + 1133.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1529

15292 = 4562 + 4572 + 4582 + 4592 + 4602 + 4622 + 4632 + 4642 + 4652 + 4662.

1265k + 1529k + 3817k + 5489k are squares for k = 1,2,3 (1102, 69742, 4760142).
1276k + 1529k + 1738k + 5258k are squares for k = 1,2,3 (992, 58852, 3953072).

Page of Squares : First Upload February 5, 2007 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1530

The integral triangle of sides 2312, 3593, 5655 has square area 15302.

15302 = (12 + 9)(122 + 9)(392 + 9) = (12 + 9)(32 + 9)(1142 + 9) = (12 + 9)(52 + 9)(62 + 9)(122 + 9)
= (32 + 9)(122 + 9)(292 + 9) = (32 + 9)(42 + 9)(52 + 9)(122 + 9) = (52 + 9)(122 + 9)(212 + 9)
= (52 + 9)(62 + 9)(392 + 9).

15302 = (1)(2)(3)(4 + 5)(6 + 7 + 8 + ... + 294),
15302 = (1)(2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 161),
15302 = (1)(2)(3)(4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 59),
15302 = (1)(2)(3)(4 + 5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 40),
15302 = (1)(2)(3)(4 + 5 + 6)(7 + 8 + 9 + ... + 23)(24 + 25 + 26 + ... + 27),
15302 = (1)(2)(3 + 4 + 5 + ... + 14)(15 + 16 + 17 + 18 + 19)(20 + 21 + 22 + ... + 25),
15302 = (1)(2)(3 + 4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 93),
15302 = (1)(2 + 3)(4 + 5 + 6 + ... + 30)(31 + 32 + 33 + ... + 54),
15302 = (1)(2 + 3 + 4 + ... + 7)(8 + 9)(10 + 11 + 12 + 13 + 14)(15 + 16 + 17 + 18 + 19),
15302 = (1 + 2)(3)(4 + 5 + 6 + ... + 20)(21 + 22 + 23 + ... + 54),
15302 = (1 + 2)(3 + 4 + 5 + ... + 14)(15)(16 + 17 + 18 + ... + 35),
15302 = (1 + 2)(3 + 4 + 5 + ... + 42)(43 + 44 + 45 + ... + 59),
15302 = (1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 76),
15302 = (1 + 2 + 3 + ... + 17)(18 + 19 + 20 + ... + 22)(23 + 24 + 25 + ... + 28),
15302 = (1 + 2 + 3 + 4 + 5)(6)(7 + 8)(9 + 10 + 11 + ... + 59),
15302 = (1 + 2 + 3 + 4 + 5)(6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 40),
15302 = (1 + 2 + 3 + 4 + 5)(6)(7 + 8 + 9 + ... + 23)(24 + 25 + 26 + 27),
15302 = (1 + 2 + 3)(4 + 5)(6 + 7 + 8 + ... + 294),
15302 = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 161),
15302 = (1 + 2 + 3)(4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 59),
15302 = (1 + 2 + 3)(4 + 5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 40),
15302 = (1 + 2 + 3)(4 + 5 + 6)(7 + 8 + 9 + ... + 23)(24 + 25 + 26 + 27).

15302 = (12)(22 + 32 + ... + 52)(62 + 72)(82 + 92 + ... + 122).

Page of Squares : First Upload February 5, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1532

15323 = 3595640768, and 352 + 92 + 52 + 62 + 42 + 02 + 72 + 62 + 82 = 1532.

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

1533

15332 = 563 + 813 + 1183.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1535

(15352 + 3) = (22 + 3)(42 + 3)(102 + 3)(132 + 3).

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1536

15362 = 643 + 1283.

402 + 1536 = 562, 402 - 1536 = 82.

Komachi equation: 15362 = 12 * 22 / 32 * 42 * 562 / 72 * 82 * 92.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 26, 2011
by Yoshio Mimura, Kobe, Japan

1538

(15382 - 4) = (32 - 4)(62 - 4)(92 - 4)(142 - 4).

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1539

15392 = 183 + 603 + 1293.

15392 = (1 + 2)(3)(4 + 5 + 6 + ... + 725).

Page of Squares : First Upload February 5, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1540

A cubic polynomial :
(X + 5672)(X + 12002)(X + 15402) = X3 + 20332X2 + 21541802X + 10478160002.

15402 = (1)(2 + 3)(4 + 5 + 6 + ... + 10)(11)(12 + 13 + 14 + ... + 43),
15402 = (1)(2 + 3 + 4 + ... + 12)(13 + 14 + ... + 19)(20 + 21 + ... + 30),
15402 = (1)(2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10 + ... + 14)(15 + 16 + 17 + ... + 25),
15402 = (1)(2 + 3 + 4 + ... + 8)(9 + 10 + 11 + ... + 19)(20 + 21 + 22 + ... + 35),
15402 = (1 + 2 + 3 + 4)(5)(6 + 7 + 8 + ... + 16)(17 + 18 + ... + 32),
15402 = (13 + 23 + ... + 553).

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1541

15412 = 2374681, a cubic with different digits.

204953k + 311282k + 893780k + 964666k are squares for k = 1,2,3 (15412, 13668672, 12847024212).

Page of Squares : First Upload February 5, 2007 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1542

15422 = 593 + 973 + 1083.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1544

15442 = 2383936, a square pegged by 3.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1546

15462 = 3042 + 3052 + 3062 + 3072 + 3082 + 3093 + 3102 + ... + 3272.

146k + 154k + 654k + 1546k are squares for k = 1,2,3 (502, 16922, 631002).

Page of Squares : First Upload February 5, 2007 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1547

A cubic polynomial :
(X + 6002)(X + 14402)(X + 15472) = X3 + 21972X2 + 25633202X + 13366080002.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1548

15482 = 163 + 1023 + 1103.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1550

15502 = (22 + 6)(52 + 6)(882 + 6).

Page of Squares : First Upload December 14, 2013 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1551

15512 = 13 + 743 + 1263.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1552

15522 = 173 + 473 + 1323 = 164 + 244 + 244 + 364.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1553

1 / 1553 = 0.000643915003219, 62 + 42 + 32 + 92 + 12 + 52 + 00322 + 192 = 1553.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1554

15542± 5 are primes.

15542 = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 32)(33 + 34 + 35 + ... + 41).

1554k + 23865k + 37740k + 47730k are squares for k = 1,2,3 (3332, 653792, 132697172).

The integral triangle of sides 111, 50680, 50737 has square area 15542.

Page of Squares : First Upload February 5, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1557

15572 = 2424249, a square pegged by 4 with just 3 kinds of digits 2, 4 and 9.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1558

15582 = 2427364, a zigzag square.

(15582 - 2) = (32 - 2)(72 - 2)(82 - 2)(112 - 2).

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1559

1559 is the first prime for which the Legendre symbol (a/p) = 1 for a = 1, 2,...,16.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1560

15602, 22952, 59842, 15602 + 22952, 22952 + 59842, 59842 + 15602 are squares.

15602 = 483 + 653 + 1273.

The integral triangle of sides 1921, 2704, 3825 (or 507, 21152, 21605) has square area 15602.

Komachi equations:
15602 = 12 * 22 / 32 * 42 * 52 / 62 * 782 * 92 = 12 / 22 * 32 * 42 * 52 * 62 * 782 / 92
 = 12 * 22 / 32 * 452 * 62 * 782 / 92.

15602 = (1)(2)(3)(4)(5)(6)(7 + 8 + ... + 19)(20)
15602 = (1)(2)(3)(4)(5)(6 + 7)(8)(9 + 10 + ... + 21)
15602 = (1)(2)(3)(4 + 5 + ... + 9)(10)(11 + 12 + ... + 15)(16)
15602 = (1)(2)(3 + 4 + ... + 10)(11 + 12 + 13)(14 + 15 + ... + 38)
15602 = (1)(2 + 3)(4)(5 + 6 + ... + 8)(9 + 10 + 11)(12)(13)
15602 = (1 + 2 + ... + 15)(16 + 17 + ... + 23)(24 + 25 + ... + 28)
15602 = (1 + 2 + ... + 4)(5 + 6 + ... + 8)(9 + 10 + 11)(12 + 13 + ... + 27)
15602 = (1 + 2 + 3 + 4 + 5)(6 + 7)(8)(9 + 10 + 11 + ... + 56),
15602 = (1 + 2 + 3)(4)(5)(6)(7 + 8 + 9 + ... + 19)(20),
15602 = (1 + 2 + 3)(4)(5)(6 + 7)(8)(9 + 10 + 11 + ... + 21),
15602 = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14 + 15)(16).

Page of Squares : First Upload February 5, 2007 ; Last Revised October 4, 2011
by Yoshio Mimura, Kobe, Japan

1561

15622 = (22 + 3)(5902 + 3).

A cubic polynomial :
(X + 2642)(X + 8642)(X + 12732) = X3 + 15612X2 + 11724722X + 2903662082.

Page of Squares : First Upload February 5, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1562

15622 = 233 + 783 + 1253.

89034k + 576378k + 866910k + 907522k are squares for k = 1,2,3 (15622, 13839322, 12613993482).

Page of Squares : First Upload July 10, 2008 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1563

15632± 2 are primes.

Komachi equations: 15632 = 9 + 87 * 65 * 432 * 1 = 9 + 87 * 65 * 432 / 1.

Page of Squares : First Upload January 18, 2010 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

1564

42 + 172 + 302 + 432 + 562 + 692 + 822 + ... + 15642 = 99662.

26k + 338k + 910k + 1226k are squares for k = 1,2,3 (502, 15642, 513322).

Page of Squares : First Upload February 5, 2007 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1566

15662 = (1 + 2)(3 + 4 + 5 + ... + 26)(27)(28 + 29 + 30).

15662 = (12 + 5)(22 + 5)(132 + 5)(162 + 5) = (22 + 5)(72 + 5)(712 + 5) = (72 + 5)(132 + 5)(162 + 5).

Komachi equations:
15662 = 92 * 872 * 62 * 52 * 42 / 32 / 22 / 102 = 92 * 872 * 62 / 52 / 42 / 32 * 22 * 102
 = 92 * 872 / 62 * 52 * 42 * 32 * 22 / 102 = 92 * 872 / 62 / 52 * 42 * 32 / 22 * 102.

Page of Squares : First Upload February 5, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1567

15672 = 363 + 643 + 1293.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1568

A cubic polynomial :
(X + 14162)(X + 15682)(X + 25832) = X3 + 33372X2 + 58915922 + 57350039042.

15682 = 284 + 284 + 284 + 284.

Page of Squares : First Upload February 5, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1569

15692 = 2461761, 2 * 46 * 17 + 6 - 1 = 1569.

Komachi equation: 15692 = 123 + 33 * 453 - 63 - 73 - 83 + 93.

Page of Squares : First Upload February 5, 2007 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1570

15702 = 472 + 492 + 512 + 532 + 542 + 552 + 562 + ... + 2452.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1571

15712 = 2468041, 2601 = 512, 484 = 222 (the 4th mosaic square).

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1572

15722 = 124 + 164 + 324 + 344.

106k + 266k + 814k + 1314k are squares for k = 1,2,3 (502, 15722, 531802).

Page of Squares : First Upload July 10, 2008 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1574

15742 = 35 + 45 + 65 + 165 + 175.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1575

15752, 16722, 91202, 15752 + 16722, 16722 + 91202, 91202 + 15752 are squares.

15752 = 104 + 204 + 304 + 354 = 154 + 304 + 304 + 304.

Komachi equations:
15752 = 92 * 82 * 72 / 62 * 52 / 42 * 32 / 22 * 102 = 92 / 82 * 72 * 62 * 52 * 42 / 32 / 22 * 102
 = 92 / 82 * 72 / 62 * 52 * 42 * 32 * 22 * 102 = 982 / 72 * 62 * 52 / 42 * 32 / 22 * 102.

15752 = (1)(2 + 3)(4 + 5 + 6 + ... + 21)(22 + 23)(24 + 25),
15752 = (1)(2 + 3)(4 + 5 + 6)(7)(8 + 9 + 10 + ... + 97),
15752 = (1)(2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 24)(25),
15752 = (1 + 2)(3 + 4)(5)(6 + 7 + 8 + ... + 19)(20 + 21 + 22 + ... + 25),
15752 = (1 + 2)(3 + 4 + 5 + ... + 12)(13 + 14 + 15 + 16 + 17)(18 + 19 + 20 + ... + 24),
15752 = (1 + 2 + 3 + ... + 14)(15 + 16 + 17 + ... + 20)(21 + 22 + 23 + ... + 29).

Page of Squares : First Upload February 5, 2007 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1581

15812 = 603 + 803 + 1213.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1584

15842 = (12 + 8)(42 + 8)(62 + 8)(162 + 8) = (22 + 8)(162 + 8)(282 + 8)
= (22 + 8)(42 + 8)(52 + 8)(162 + 8) = (42 + 8)(52 + 8)(62 + 8)(82 + 8) = (42 + 8)(82 + 8)(382 + 8)
= (62 + 8)(82 + 8)(282 + 8) = (72 - 1)(102 - 1)(232 - 1).

Cubic polynomials :
(X + 3482)(X + 5392)(X + 15842) = X3 + 17092X2 + 10334282X + 2971140482,
(X + 4482)(X + 15842)(X + 233312) = X3 + 233892 + 384125282X + 16556424192.

15842 = (1)(2)(3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 108),
15842 = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + ... + 27)(28 + 29 + 30 + ... + 36),
15842 = (1 + 2 + 3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 108).

Page of Squares : First Upload February 5, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1585

15852 = 2512225, a square with just 3 kinds of digits 1, 2 and 5.

430k + 740k + 1470k + 1585k are squares for k = 1,2,3 (652, 23252, 874252).

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
1585 - 7450 - 7976 - 12017 - ... - 2960 - 4441 - 3617 - 1585
(Note f(1585) = 152 + 852 = 7450,   f(7450) = 742 + 502 = 7976, etc. See 41)

Page of Squares : First Upload February 5, 2007 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1586

15862 = 2515396, a zigzag square.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1588

15882 = 533 + 803 + 1233.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1589

1589 = (12 + 22 + 32 + ... + 2272) / (12 + 22 + 32 + ... + 192).

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

1592

15922 = 13 + 423 + 1353.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1593

15932 = 2537649, a square with different digits.

15932 = 183 + 813 + 1263 = 214 + 244 + 244 + 364.

Page of Squares : First Upload February 5, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1594

15942 = 2540836, a square with different digits.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan

1595

1595 = (12 + 22 + 32 + ... + 872) / (12 + 22 + 32 + ... + 72).

15952 = 2452 + 2472 + 2492 + 2512 + 2532 + 2552 + 2572 + ... + 3092.

137170k + 283910k + 880440k + 1242505k are squares for k = 1,2,3 (15952, 15551252, 16205439252).

Page of Squares : First Upload February 5, 2007 ; Last Revised April 26, 2011
by Yoshio Mimura, Kobe, Japan

1596

15962 = (12 + 3)(42 + 3)(122 + 3)(152 + 3) = (22 + 3)(42 + 3)(92 + 3)(152 + 3).

15962 + 15972 + 15982 + ... + 16242 = 16252 + 16262 + 16272 + ... + 16522.

Komachi equation: 15962 = 122 * 32 * 4562 * 72 / 82 / 92.

15962 = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 13)(14)(15 + 16 + 17 + ... + 23),
15962 = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 13)(14 + 15 + 16 + ... + 70).

15962 = 13 + 23 + 33 + 43 + 53 + ... + 563.

Page of Squares : First Upload February 5, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1598

1 / 1598 = 0.000625..., and 25 = 52.

15982 = 2553604 appears in the decimal expressions of e:
  e = 2.71828•••2553604••• (from the 8378th digit)
  (2553604 is the fourth 7-digit square in the expression of e.)

Page of Squares : First Upload February 5, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1599

15992 = 3012 + 3022 + 3032 + 3042 + 3052 + 3062 + 3072 + ... + 3262.

1 / 1599 = 0.000625..., and 625 = 252.

Page of Squares : First Upload February 5, 2007 ; Last Revised February 5, 2007
by Yoshio Mimura, Kobe, Japan