## 1500

1500^{2} = 5^{3} + 90^{3} + 115^{3}.

1500^{2} = (1)(2)(3 + 4 + 5 + ... + 22)(23 + 24 + 25 + ... + 97),

1500^{2} = (1)(2 + 3)(4 + 5 + 6 + ... + 11)(12)(13 + 14 + 15 + ... + 37).

1500^{2} = (1)(2 + 3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 112).

1500^{2} = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 17)(18 + 19 + 20 + ... + 22).

by Yoshio Mimura, Kobe, Japan

## 1501

A cubic polynomial :

(X + 704^{2})(X + 861^{2})(X + 1008^{2}) = X^{3} + 1501^{2}X^{2} + 1274448^{2}X + 610993152^{2}.

1501^{2} = 2253001,

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

by Yoshio Mimura, Kobe, Japan

## 1502

1502^{2} = 2256004, 2 + 2 * 5 * 600 / 4 = 2 + 25 * 60 + 0 * 4 = 1502.

by Yoshio Mimura, Kobe, Japan

## 1503

1503^{2} = 12^{3} + 48^{3} + 129^{3}.

by Yoshio Mimura, Kobe, Japan

## 1504

1504^{2} = 56^{3} + 88^{3} + 112^{3} = 64^{3} + 76^{3} + 116^{3}.

A cubic polynomial :

(X + 1504^{2})(X + 2688^{2})(X + 3807^{2}) = X^{3} + 4897^{2} + 12403488^{2}X + 15390756864^{2}.

1504^{2} = 2262016 appears in the decimal expressions of e:

e = 2.71828•••2262016••• (from the 72805th digit)

by Yoshio Mimura, Kobe, Japan

## 1505

1505^{k} + 4042^{k} + 4988^{k} + 6106^{k} are squares for k = 1,2,3 (129^{2}, 8987^{2}, 648999^{2}).

344^{k} + 1505^{k} + 4558^{k} + 10234^{k} are squares for k = 1,2,3 (129^{2}, 11309^{2}, 1081665^{2}).

by Yoshio Mimura, Kobe, Japan

## 1507

1507^{2} = 8^{5} + 9^{5} + 15^{5} + 17^{5}.

292358^{k} + 304414^{k} + 739937^{k} + 934340^{k} are squares for k = 1,2,3 (1507^{2}, 1264373^{2}, 1128711353^{2}).

by Yoshio Mimura, Kobe, Japan

## 1509

75450^{k} + 202206^{k} + 683577^{k} + 1315848^{k} are squares for k = 1,2,3 (1509^{2}, 1498437^{2}, 1614450429^{2}).

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, 2007by Yoshio Mimura, Kobe, Japan

## 1512

1512^{2}± 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.

1512^{2} = 2286144, 228 * 6 + 144 = 1512.

Komachi equation: 1512^{2} = 1^{2} * 234^{2} * 56^{2} / 78^{2} * 9^{2}.

1512^{2} = 3^{5} + 6^{5} + 15^{5} + 15^{5} + 15^{5}.

1512^{2} = (1)(2)(3)(4)(5 + 6 + 7 + ... + 436),

1512^{2} = (1)(2)(3)(4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15)(16),

1512^{2} = (1)(2)(3)(4 + 5 + 6 + ... + 24)(25 + 26 + 27 + ... + 56),

1512^{2} = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 571),

1512^{2} = (1)(2)(3 + 4)(5 + 6 + 7)(8)(9)(10 + 11 + 12 + ... + 18),

1512^{2} = (1)(2)(3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14),

1512^{2} = (1)(2)(3 + 4 + 5 + ... + 29)(30 + 31 + 32 + ... + 78),

1512^{2} = (1)(2)(3 + 4 + 5 + 6)(7)(8)(9)(10 + 11 + 12 + ... + 18),

1512^{2} = (1)(2)(3 + 4 + 5 + 6)(7)(8)(9 + 10 + 11 + 12)(13 + 14),

1512^{2} = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 14)(15 + 16 + 17 + ... + 41),

1512^{2} = (1)(2)(3 + 4 + 5 + ... + 9)(10 + 11 + 12 + ... + 233),

1512^{2} = (1)(2 + 3 + 4 + ... + 25)(26 + 27 + 28 + ... + 121),

1512^{2} = (1)(2 + 3 + 4 + 5)(6)(7)(8 + 9 + 10 + ... + 88),

1512^{2} = (1)(2 + 3 + 4 + 5)(6)(7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15),

1512^{2} = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10 + ... + 28)(29 + 30 + 31 + ... + 35),

1512^{2} = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10 + ... + 55)(56),

1512^{2} = (1)(2 + 3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14 + 15),

1512^{2} = (1 + 2)(3)(4 + 5)(6)(7)(8)(9 + 10 + 11 + ... + 15),

1512^{2} = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + ... + 26)(27 + 28 + 29),

1512^{2} = (1 + 2)(3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11 + ... + 15)(16),

1512^{2} = (1 + 2)(3 + 4 + 5 + ... + 9)(10 + 11)(12 + 13 + 14 + 15)(16),

1512^{2} = (1 + 2)(3 + 4 + 5)(6)(7)(8)(9)(10 + 11),

1512^{2} = (1 + 2)(3 + 4 + 5)(6)(7)(8 + 9 + 10 + ... + 55),

1512^{2} = (1 + 2)(3 + 4 + 5)(6)(7 + 8 + 9 + ... + 14)(15 + 16 + 17 + ... + 21),

1512^{2} = (1 + 2 + 3 + ... + 7)(8)(9)(10 + 11)(12 + 13 + 14 + 15),

1512^{2} = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 436),

1512^{2} = (1 + 2 + 3)(4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15)(16),

1512^{2} = (1 + 2 + 3)(4 + 5 + 6 + ... + 24)(25 + 26 + 27 + ... + 56).

by Yoshio Mimura, Kobe, Japan

## 1513

S_{2}(113) + S_{2}(175) = 1513^{2}, where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

1513^{2} = 2289169, 2 - 2 - 8 + 9 * 169 = 1513.

by Yoshio Mimura, Kobe, Japan

## 1515

1515^{2} = 2295225, a square with just 3 kinds of digits 2, 5 and 9.

1515^{2} = 2295225, 2 * 2 * 95 * 2 * 2 - 5 = 1515 .

by Yoshio Mimura, Kobe, Japan

## 1516

82^{k} + 182^{k} + 722^{k} + 1318^{k} are squares for k = 1,2,3 (48^{2}, 1516^{2}, 51696^{2}).

by Yoshio Mimura, Kobe, Japan

## 1517

1517^{k} + 2257^{k} + 12173^{k} + 33337^{k} are squares for k = 1,2,3 (222^{2}, 35594^{2}, 6234426^{2}).

by Yoshio Mimura, Kobe, Japan

## 1518

The integral triangle of sides 5336, 6843, 12155 has square area 1518^{2}.

1518^{2} = 2304324, 2304 = 48^{2} and 324 = 18^{2}.

1518^{2} = (1 + 2 + 3 + ... + 11)(12 + 13 + 14 + ... + 264).

1518^{2} = (3^{3} + 9)(40^{3} + 9).

1518^{2} = 7^{2} + 8^{2} + 9^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + ... + 190^{2}.

1518^{2} = 2304324 appears in the decimal expressions of π:

π = 3.14159•••2304324••• (from the 62461st digit)

by Yoshio Mimura, Kobe, Japan

## 1519

1519^{2} = 2307361, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 1520

1520^{2} = 21^{3} + 67^{3} + 126^{3} = 64^{3} + 74^{3} + 118^{3}.

Komachi equation: 1520^{2} = 9^{2} * 8^{2} * 76^{2} / 54^{2} * 3^{2} / 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 1521

The square of 39.

1521^{2} = 2313441, a square every digit of which is non-zero and smaller than 5.

1521^{2} = (356 + 357 + 358)^{2} + (359 + 360 + 361)^{2}.

1521^{2} = 19^{3} + 89^{3} + 117^{3}.

by Yoshio Mimura, Kobe, Japan

## 1522

1522^{2} = 2316484, a zigzag square.

1522^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 1524

1 / 1524 = 0.0006561... and 6561 = 81^{2}.

1524^{2} = 2322576, 2 * 32 * 25 - 76 = 1524.

1524^{2} = 55^{3} + 9^{5} + 8^{7}.

by Yoshio Mimura, Kobe, Japan

## 1525

1525^{2} = (2^{2} + 1)(682^{2} + 1).

1525^{2} = 73^{2} + 74^{2} + 75^{2} + 76^{2} + 77^{2} + 78^{2} + 79^{2} + ... +194^{2}.

by Yoshio Mimura, Kobe, Japan

## 1526

1526^{2} = 2328676, a zigzag square.

Komachi equations:

1526^{2} = 98^{2} / 7^{2} * 654^{2} / 3^{2} / 2^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 1527

1527^{2} = 9^{3} + 100^{3} + 110^{3} = 16^{3} + 96^{3} + 113^{3}.

by Yoshio Mimura, Kobe, Japan

## 1529

1529^{2} = 456^{2} + 457^{2} + 458^{2} + 459^{2} + 460^{2} + 462^{2} + 463^{2} + 464^{2} + 465^{2} + 466^{2}.

1265^{k} + 1529^{k} + 3817^{k} + 5489^{k} are squares for k = 1,2,3 (110^{2}, 6974^{2}, 476014^{2}).

1276^{k} + 1529^{k} + 1738^{k} + 5258^{k} are squares for k = 1,2,3 (99^{2}, 5885^{2}, 395307^{2}).

by Yoshio Mimura, Kobe, Japan

## 1530

The integral triangle of sides 2312, 3593, 5655 has square area 1530^{2}.

1530^{2} = (1^{2} + 9)(12^{2} + 9)(39^{2} + 9) = (1^{2} + 9)(3^{2} + 9)(114^{2} + 9) = (1^{2} + 9)(5^{2} + 9)(6^{2} + 9)(12^{2} + 9)

= (3^{2} + 9)(12^{2} + 9)(29^{2} + 9) = (3^{2} + 9)(4^{2} + 9)(5^{2} + 9)(12^{2} + 9) = (5^{2} + 9)(12^{2} + 9)(21^{2} + 9)

= (5^{2} + 9)(6^{2} + 9)(39^{2} + 9).

1530^{2} = (1)(2)(3)(4 + 5)(6 + 7 + 8 + ... + 294),

1530^{2} = (1)(2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 161),

1530^{2} = (1)(2)(3)(4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 59),

1530^{2} = (1)(2)(3)(4 + 5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 40),

1530^{2} = (1)(2)(3)(4 + 5 + 6)(7 + 8 + 9 + ... + 23)(24 + 25 + 26 + ... + 27),

1530^{2} = (1)(2)(3 + 4 + 5 + ... + 14)(15 + 16 + 17 + 18 + 19)(20 + 21 + 22 + ... + 25),

1530^{2} = (1)(2)(3 + 4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 93),

1530^{2} = (1)(2 + 3)(4 + 5 + 6 + ... + 30)(31 + 32 + 33 + ... + 54),

1530^{2} = (1)(2 + 3 + 4 + ... + 7)(8 + 9)(10 + 11 + 12 + 13 + 14)(15 + 16 + 17 + 18 + 19),

1530^{2} = (1 + 2)(3)(4 + 5 + 6 + ... + 20)(21 + 22 + 23 + ... + 54),

1530^{2} = (1 + 2)(3 + 4 + 5 + ... + 14)(15)(16 + 17 + 18 + ... + 35),

1530^{2} = (1 + 2)(3 + 4 + 5 + ... + 42)(43 + 44 + 45 + ... + 59),

1530^{2} = (1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 76),

1530^{2} = (1 + 2 + 3 + ... + 17)(18 + 19 + 20 + ... + 22)(23 + 24 + 25 + ... + 28),

1530^{2} = (1 + 2 + 3 + 4 + 5)(6)(7 + 8)(9 + 10 + 11 + ... + 59),

1530^{2} = (1 + 2 + 3 + 4 + 5)(6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 40),

1530^{2} = (1 + 2 + 3 + 4 + 5)(6)(7 + 8 + 9 + ... + 23)(24 + 25 + 26 + 27),

1530^{2} = (1 + 2 + 3)(4 + 5)(6 + 7 + 8 + ... + 294),

1530^{2} = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 161),

1530^{2} = (1 + 2 + 3)(4 + 5 + 6)(7 + 8)(9 + 10 + 11 + ... + 59),

1530^{2} = (1 + 2 + 3)(4 + 5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 40),

1530^{2} = (1 + 2 + 3)(4 + 5 + 6)(7 + 8 + 9 + ... + 23)(24 + 25 + 26 + 27).

1530^{2} = (1^{2})(2^{2} + 3^{2} + ... + 5^{2})(6^{2} + 7^{2})(8^{2} + 9^{2} + ... + 12^{2}).

by Yoshio Mimura, Kobe, Japan

## 1532

1532^{3} = 3595640768, and 35^{2} + 9^{2} + 5^{2} + 6^{2} + 4^{2} + 0^{2} + 7^{2} + 6^{2} + 8^{2} = 1532.

by Yoshio Mimura, Kobe, Japan

## 1533

1533^{2} = 56^{3} + 81^{3} + 118^{3}.

by Yoshio Mimura, Kobe, Japan

## 1535

(1535^{2} + 3) = (2^{2} + 3)(4^{2} + 3)(10^{2} + 3)(13^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 1536

1536^{2} = 64^{3} + 128^{3}.

40^{2} + 1536 = 56^{2}, 40^{2} - 1536 = 8^{2}.

Komachi equation: 1536^{2} = 1^{2} * 2^{2} / 3^{2} * 4^{2} * 56^{2} / 7^{2} * 8^{2} * 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 1538

(1538^{2} - 4) = (3^{2} - 4)(6^{2} - 4)(9^{2} - 4)(14^{2} - 4).

by Yoshio Mimura, Kobe, Japan

## 1539

1539^{2} = 18^{3} + 60^{3} + 129^{3}.

1539^{2} = (1 + 2)(3)(4 + 5 + 6 + ... + 725).

by Yoshio Mimura, Kobe, Japan

## 1540

A cubic polynomial :

(X + 567^{2})(X + 1200^{2})(X + 1540^{2}) = X^{3} + 2033^{2}X^{2} + 2154180^{2}X + 1047816000^{2}.

1540^{2} = (1)(2 + 3)(4 + 5 + 6 + ... + 10)(11)(12 + 13 + 14 + ... + 43),

1540^{2} = (1)(2 + 3 + 4 + ... + 12)(13 + 14 + ... + 19)(20 + 21 + ... + 30),

1540^{2} = (1)(2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10 + ... + 14)(15 + 16 + 17 + ... + 25),

1540^{2} = (1)(2 + 3 + 4 + ... + 8)(9 + 10 + 11 + ... + 19)(20 + 21 + 22 + ... + 35),

1540^{2} = (1 + 2 + 3 + 4)(5)(6 + 7 + 8 + ... + 16)(17 + 18 + ... + 32),

1540^{2} = (1^{3} + 2^{3} + ... + 55^{3}).

by Yoshio Mimura, Kobe, Japan

## 1541

1541^{2} = 2374681, a cubic with different digits.

204953^{k} + 311282^{k} + 893780^{k} + 964666^{k} are squares for k = 1,2,3 (1541^{2}, 1366867^{2}, 1284702421^{2}).

by Yoshio Mimura, Kobe, Japan

## 1542

1542^{2} = 59^{3} + 97^{3} + 108^{3}.

by Yoshio Mimura, Kobe, Japan

## 1544

1544^{2} = 2383936, a square pegged by 3.

by Yoshio Mimura, Kobe, Japan

## 1546

1546^{2} = 304^{2} + 305^{2} + 306^{2} + 307^{2} + 308^{2} + 309^{3} + 310^{2} + ... + 327^{2}.

146^{k} + 154^{k} + 654^{k} + 1546^{k} are squares for k = 1,2,3 (50^{2}, 1692^{2}, 63100^{2}).

by Yoshio Mimura, Kobe, Japan

## 1547

A cubic polynomial :

(X + 600^{2})(X + 1440^{2})(X + 1547^{2}) = X^{3} + 2197^{2}X^{2} + 2563320^{2}X + 1336608000^{2}.

by Yoshio Mimura, Kobe, Japan

## 1548

1548^{2} = 16^{3} + 102^{3} + 110^{3}.

by Yoshio Mimura, Kobe, Japan

## 1550

1550^{2} = (2^{2} + 6)(5^{2} + 6)(88^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 1551

1551^{2} = 1^{3} + 74^{3} + 126^{3}.

by Yoshio Mimura, Kobe, Japan

## 1552

1552^{2} = 17^{3} + 47^{3} + 132^{3} = 16^{4} + 24^{4} + 24^{4} + 36^{4}.

by Yoshio Mimura, Kobe, Japan

## 1553

1 / 1553 = 0.000643915003219, 6^{2} + 4^{2} + 3^{2} + 9^{2} + 1^{2} + 5^{2} + 0032^{2} + 19^{2} = 1553.

by Yoshio Mimura, Kobe, Japan

## 1554

1554^{2}± 5 are primes.

1554^{2} = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 32)(33 + 34 + 35 + ... + 41).

1554^{k} + 23865^{k} + 37740^{k} + 47730^{k} are squares for k = 1,2,3 (333^{2}, 65379^{2}, 13269717^{2}).

The integral triangle of sides 111, 50680, 50737 has square area 1554^{2}.

by Yoshio Mimura, Kobe, Japan

## 1557

1557^{2} = 2424249, a square pegged by 4 with just 3 kinds of digits 2, 4 and 9.

by Yoshio Mimura, Kobe, Japan

## 1558

1558^{2} = 2427364, a zigzag square.

(1558^{2} - 2) = (3^{2} - 2)(7^{2} - 2)(8^{2} - 2)(11^{2} - 2).

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, 2007by Yoshio Mimura, Kobe, Japan

## 1560

1560^{2}, 2295^{2}, 5984^{2}, 1560^{2} + 2295^{2}, 2295^{2} + 5984^{2}, 5984^{2} + 1560^{2} are squares.

1560^{2} = 48^{3} + 65^{3} + 127^{3}.

The integral triangle of sides 1921, 2704, 3825 (or 507, 21152, 21605) has square area 1560^{2}.

Komachi equations:

1560^{2} = 1^{2} * 2^{2} / 3^{2} * 4^{2} * 5^{2} / 6^{2} * 78^{2} * 9^{2} = 1^{2} / 2^{2} * 3^{2} * 4^{2} * 5^{2} * 6^{2} * 78^{2} / 9^{2}

= 1^{2} * 2^{2} / 3^{2} * 45^{2} * 6^{2} * 78^{2} / 9^{2}.

1560^{2} = (1)(2)(3)(4)(5)(6)(7 + 8 + ... + 19)(20)

1560^{2} = (1)(2)(3)(4)(5)(6 + 7)(8)(9 + 10 + ... + 21)

1560^{2} = (1)(2)(3)(4 + 5 + ... + 9)(10)(11 + 12 + ... + 15)(16)

1560^{2} = (1)(2)(3 + 4 + ... + 10)(11 + 12 + 13)(14 + 15 + ... + 38)

1560^{2} = (1)(2 + 3)(4)(5 + 6 + ... + 8)(9 + 10 + 11)(12)(13)

1560^{2} = (1 + 2 + ... + 15)(16 + 17 + ... + 23)(24 + 25 + ... + 28)

1560^{2} = (1 + 2 + ... + 4)(5 + 6 + ... + 8)(9 + 10 + 11)(12 + 13 + ... + 27)

1560^{2} = (1 + 2 + 3 + 4 + 5)(6 + 7)(8)(9 + 10 + 11 + ... + 56),

1560^{2} = (1 + 2 + 3)(4)(5)(6)(7 + 8 + 9 + ... + 19)(20),

1560^{2} = (1 + 2 + 3)(4)(5)(6 + 7)(8)(9 + 10 + 11 + ... + 21),

1560^{2} = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14 + 15)(16).

by Yoshio Mimura, Kobe, Japan

## 1561

1562^{2} = (2^{2} + 3)(590^{2} + 3).

A cubic polynomial :

(X + 264^{2})(X + 864^{2})(X + 1273^{2}) = X^{3} + 1561^{2}X^{2} + 1172472^{2}X + 290366208^{2}.

by Yoshio Mimura, Kobe, Japan

## 1562

1562^{2} = 23^{3} + 78^{3} + 125^{3}.

89034^{k} + 576378^{k} + 866910^{k} + 907522^{k} are squares for k = 1,2,3 (1562^{2}, 1383932^{2}, 1261399348^{2}).

by Yoshio Mimura, Kobe, Japan

## 1563

1563^{2}± 2 are primes.

Komachi equations: 1563^{2} = 9 + 87 * 65 * 432 * 1 = 9 + 87 * 65 * 432 / 1.

by Yoshio Mimura, Kobe, Japan

## 1564

4^{2} + 17^{2} + 30^{2} + 43^{2} + 56^{2} + 69^{2} + 82^{2} + ... + 1564^{2} = 9966^{2}.

26^{k} + 338^{k} + 910^{k} + 1226^{k} are squares for k = 1,2,3 (50^{2}, 1564^{2}, 51332^{2}).

by Yoshio Mimura, Kobe, Japan

## 1566

1566^{2} = (1 + 2)(3 + 4 + 5 + ... + 26)(27)(28 + 29 + 30).

1566^{2} = (1^{2} + 5)(2^{2} + 5)(13^{2} + 5)(16^{2} + 5) = (2^{2} + 5)(7^{2} + 5)(71^{2} + 5) = (7^{2} + 5)(13^{2} + 5)(16^{2} + 5).

Komachi equations:

1566^{2} = 9^{2} * 87^{2} * 6^{2} * 5^{2} * 4^{2} / 3^{2} / 2^{2} / 10^{2} = 9^{2} * 87^{2} * 6^{2} / 5^{2} / 4^{2} / 3^{2} * 2^{2} * 10^{2}

= 9^{2} * 87^{2} / 6^{2} * 5^{2} * 4^{2} * 3^{2} * 2^{2} / 10^{2} = 9^{2} * 87^{2} / 6^{2} / 5^{2} * 4^{2} * 3^{2} / 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 1567

1567^{2} = 36^{3} + 64^{3} + 129^{3}.

by Yoshio Mimura, Kobe, Japan

## 1568

A cubic polynomial :

(X + 1416^{2})(X + 1568^{2})(X + 2583^{2}) = X^{3} + 3337^{2}X^{2} + 5891592^{2} + 5735003904^{2}.

1568^{2} = 28^{4} + 28^{4} + 28^{4} + 28^{4}.

by Yoshio Mimura, Kobe, Japan

## 1569

1569^{2} = 2461761, 2 * 46 * 17 + 6 - 1 = 1569.

Komachi equation: 1569^{2} = 12^{3} + 3^{3} * 45^{3} - 6^{3} - 7^{3} - 8^{3} + 9^{3}.

by Yoshio Mimura, Kobe, Japan

## 1570

1570^{2} = 47^{2} + 49^{2} + 51^{2} + 53^{2} + 54^{2} + 55^{2} + 56^{2} + ... + 245^{2}.

by Yoshio Mimura, Kobe, Japan

## 1571

1571^{2} = 2468041, 2601 = 51^{2}, 484 = 22^{2} (the 4th mosaic square).

by Yoshio Mimura, Kobe, Japan

## 1572

1572^{2} = 12^{4} + 16^{4} + 32^{4} + 34^{4}.

106^{k} + 266^{k} + 814^{k} + 1314^{k} are squares for k = 1,2,3 (50^{2}, 1572^{2}, 53180^{2}).

by Yoshio Mimura, Kobe, Japan

## 1574

1574^{2} = 3^{5} + 4^{5} + 6^{5} + 16^{5} + 17^{5}.

by Yoshio Mimura, Kobe, Japan

## 1575

1575^{2}, 1672^{2}, 9120^{2}, 1575^{2} + 1672^{2}, 1672^{2} + 9120^{2}, 9120^{2} + 1575^{2} are squares.

1575^{2} = 10^{4} + 20^{4} + 30^{4} + 35^{4} = 15^{4} + 30^{4} + 30^{4} + 30^{4}.

Komachi equations:

1575^{2} = 9^{2} * 8^{2} * 7^{2} / 6^{2} * 5^{2} / 4^{2} * 3^{2} / 2^{2} * 10^{2} = 9^{2} / 8^{2} * 7^{2} * 6^{2} * 5^{2} * 4^{2} / 3^{2} / 2^{2} * 10^{2}

= 9^{2} / 8^{2} * 7^{2} / 6^{2} * 5^{2} * 4^{2} * 3^{2} * 2^{2} * 10^{2} = 98^{2} / 7^{2} * 6^{2} * 5^{2} / 4^{2} * 3^{2} / 2^{2} * 10^{2}.

1575^{2} = (1)(2 + 3)(4 + 5 + 6 + ... + 21)(22 + 23)(24 + 25),

1575^{2} = (1)(2 + 3)(4 + 5 + 6)(7)(8 + 9 + 10 + ... + 97),

1575^{2} = (1)(2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 24)(25),

1575^{2} = (1 + 2)(3 + 4)(5)(6 + 7 + 8 + ... + 19)(20 + 21 + 22 + ... + 25),

1575^{2} = (1 + 2)(3 + 4 + 5 + ... + 12)(13 + 14 + 15 + 16 + 17)(18 + 19 + 20 + ... + 24),

1575^{2} = (1 + 2 + 3 + ... + 14)(15 + 16 + 17 + ... + 20)(21 + 22 + 23 + ... + 29).

by Yoshio Mimura, Kobe, Japan

## 1581

1581^{2} = 60^{3} + 80^{3} + 121^{3}.

by Yoshio Mimura, Kobe, Japan

## 1584

1584^{2} = (1^{2} + 8)(4^{2} + 8)(6^{2} + 8)(16^{2} + 8) = (2^{2} + 8)(16^{2} + 8)(28^{2} + 8)

= (2^{2} + 8)(4^{2} + 8)(5^{2} + 8)(16^{2} + 8) = (4^{2} + 8)(5^{2} + 8)(6^{2} + 8)(8^{2} + 8) = (4^{2} + 8)(8^{2} + 8)(38^{2} + 8)

= (6^{2} + 8)(8^{2} + 8)(28^{2} + 8) = (7^{2} - 1)(10^{2} - 1)(23^{2} - 1).

Cubic polynomials :

(X + 348^{2})(X + 539^{2})(X + 1584^{2}) = X^{3} + 1709^{2}X^{2} + 1033428^{2}X + 297114048^{2},

(X + 448^{2})(X + 1584^{2})(X + 23331^{2}) = X^{3} + 23389^{2} + 38412528^{2}X + 16556424192.

1584^{2} = (1)(2)(3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 108),

1584^{2} = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + ... + 27)(28 + 29 + 30 + ... + 36),

1584^{2} = (1 + 2 + 3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 108).

by Yoshio Mimura, Kobe, Japan

## 1585

1585^{2} = 2512225, a square with just 3 kinds of digits 1, 2 and 5.

430^{k} + 740^{k} + 1470^{k} + 1585^{k} are squares for k = 1,2,3 (65^{2}, 2325^{2}, 87425^{2}).

Loop of length 56 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

1585 - 7450 - 7976 - 12017 - ... - 2960 - 4441 - 3617 - 1585

(Note f(1585) = 15^{2} + 85^{2} = 7450, f(7450) = 74^{2} + 50^{2} = 7976, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 1586

1586^{2} = 2515396, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 1588

1588^{2} = 53^{3} + 80^{3} + 123^{3}.

by Yoshio Mimura, Kobe, Japan

## 1589

1589 = (1^{2} + 2^{2} + 3^{2} + ... + 227^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 19^{2}).

by Yoshio Mimura, Kobe, Japan

## 1592

1592^{2} = 1^{3} + 42^{3} + 135^{3}.

by Yoshio Mimura, Kobe, Japan

## 1593

1593^{2} = 2537649, a square with different digits.

1593^{2} = 18^{3} + 81^{3} + 126^{3} = 21^{4} + 24^{4} + 24^{4} + 36^{4}.

by Yoshio Mimura, Kobe, Japan

## 1594

1594^{2} = 2540836, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 1595

1595 = (1^{2} + 2^{2} + 3^{2} + ... + 87^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 7^{2}).

1595^{2} = 245^{2} + 247^{2} + 249^{2} + 251^{2} + 253^{2} + 255^{2} + 257^{2} + ... + 309^{2}.

137170^{k} + 283910^{k} + 880440^{k} + 1242505^{k} are squares for k = 1,2,3 (1595^{2}, 1555125^{2}, 1620543925^{2}).

by Yoshio Mimura, Kobe, Japan

## 1596

1596^{2} = (1^{2} + 3)(4^{2} + 3)(12^{2} + 3)(15^{2} + 3) = (2^{2} + 3)(4^{2} + 3)(9^{2} + 3)(15^{2} + 3).

1596^{2} + 1597^{2} + 1598^{2} + ... + 1624^{2} = 1625^{2} + 1626^{2} + 1627^{2} + ... + 1652^{2}.

Komachi equation: 1596^{2} = 12^{2} * 3^{2} * 456^{2} * 7^{2} / 8^{2} / 9^{2}.

1596^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 13)(14)(15 + 16 + 17 + ... + 23),

1596^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 13)(14 + 15 + 16 + ... + 70).

1596^{2} = 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + ... + 56^{3}.

by Yoshio Mimura, Kobe, Japan

## 1598

1 / 1598 = 0.000625..., and 25 = 5^{2}.

1598^{2} = 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.)

by Yoshio Mimura, Kobe, Japan

## 1599

1599^{2} = 301^{2} + 302^{2} + 303^{2} + 304^{2} + 305^{2} + 306^{2} + 307^{2} + ... + 326^{2}.

1 / 1599 = 0.000625..., and 625 = 25^{2}.

by Yoshio Mimura, Kobe, Japan