## 5700

5700^{2} = 40^{3} + 222^{3} + 278^{3} = 118^{3} + 209^{3} + 279^{3}.

5700^{2} = 32490000, and 324 = 18^{2} and 90000 = 300^{2}.

228^{k} + 4104^{k} + 5700^{k} + 7657^{k} are squares for k = 1,2,3 (133^{2}, 10393^{2}, 838603^{2}).

by Yoshio Mimura, Kobe, Japan

## 5705

5705^{2} = 47^{3} + 247^{3} + 259^{3}.

by Yoshio Mimura, Kobe, Japan

## 5708

1 / 5708 = 0.00017519, 1^{2} + 75^{2} + 1^{2} + 9^{2} = 5708.

by Yoshio Mimura, Kobe, Japan

## 5709

1122^{k} + 1353^{k} + 2508^{k} + 4818^{k} are squares for k = 1,2,3 (99^{2}, 5709^{2}, 362637^{2}).

by Yoshio Mimura, Kobe, Japan

## 5711

The square root of 5711 is 75.571..., and 75 = 5^{2} + 7^{2} + 1^{2}.

5711 is the first prime for which the Legendre symbol (a / 5711) = 1 for a = 1, 2,..., 18.

(5711 is the 3rd prime for which the Legendre symbol (a / 5711) = 1 for a = 1, 2,..., 16).

by Yoshio Mimura, Kobe, Japan

## 5712

5712^{2} = 128^{3} + 244^{3} + 252^{3}.

by Yoshio Mimura, Kobe, Japan

## 5713

Loop of length 10 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

5713 - 3418 - 1480 - 6596 - ... - 1268 - 4768 - 6833 - 5713

(Note f(5713) = 57^{2} + 13^{2} = 3418, f(3418) = 34^{2} + 18^{2} = 1480, etc. See 1268)

by Yoshio Mimura, Kobe, Japan

## 5714

5714^{2} = 32649796, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5719

S_{2}(5719) = S_{2}(4) x S_{2}(21) x S_{2}(123).

by Yoshio Mimura, Kobe, Japan

## 5720

5720^{2} = 185 x 186 + 186 x 187 + 187 x 188 + 188 x 189 + ... + 470 x 471.

by Yoshio Mimura, Kobe, Japan

## 5723

5723^{2} = 32752729, 3^{2} + 2^{2} + 7^{2} + 5^{2} + 2^{2} + 7^{2} + 2^{2} + 9^{2} = 15^{2}.

5723^{2} = 32752729, 3275 + 272 * 9 = 5723.

by Yoshio Mimura, Kobe, Japan

## 5727

5727^{2} = 109^{3} + 237^{3} + 263^{3}.

by Yoshio Mimura, Kobe, Japan

## 5728

5728^{2} = 32809984, 32 * 80 + 99 * 8 * 4 = 5728.

by Yoshio Mimura, Kobe, Japan

## 5732

5732^{2} = 66^{3} + 130^{3} + 312^{3}.

by Yoshio Mimura, Kobe, Japan

## 5733

5733^{5} = 6193123048039183893 : 61^{2} + 9^{2} + 3^{2} + 12^{2} + 3^{2} + 0^{2} + 4^{2} + 8^{2} + 0^{2} + 3^{2} + 9^{2} + 1^{2} + 8^{2} + 38^{2} + 9^{2} + 3^{2} = 5733.

by Yoshio Mimura, Kobe, Japan

## 5734

5734^{2} = 1^{3} + 158^{3} + 307^{3}.

by Yoshio Mimura, Kobe, Japan

## 5735

5735^{5} = 6203933174657834375 : 6^{2} + 2^{2} + 0^{2} + 3^{2} + 9^{2} + 3^{2} + 3^{2} + 1^{2} + 7^{2} + 46^{2} + 57^{2} + 8^{2} + 3^{2} + 4^{2} + 3^{2} + 7^{2} + 5^{2} = 5735.

by Yoshio Mimura, Kobe, Japan

## 5737

5737^{5} = 6214758391312632457 : 6^{2} + 2^{2} + 1^{2} + 47^{2} + 5^{2} + 8^{2} + 3^{2} + 9^{2} + 13^{2} + 1^{2} + 2^{2} + 6^{2} + 32^{2} + 45^{2} + 7^{2} = 6^{2} + 21^{2} + 4^{2} + 7^{2} + 58^{2} + 3^{2} + 9^{2} + 1^{2} + 31^{2} + 26^{2} + 3^{2} + 2^{2} + 4^{2} + 5^{2} + 7^{2} = 5737.

by Yoshio Mimura, Kobe, Japan

## 5739

5739^{2} = 1^{3} + 92^{3} + 318^{3}.

5739^{2} = 32936121, a zigzag square.

1 / 5739 = 0.00017424, and 17424 = 132^{2}.

by Yoshio Mimura, Kobe, Japan

## 5740

Komachi equations:

5740^{2} = 123^{2} / 4^{2} * 5^{2} * 6^{2} * 7^{2} * 8^{2} / 9^{2} = 123^{2} * 45^{2} / 6^{2} * 7^{2} * 8^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 5741

5741^{2} = 67^{3} + 155^{3} + 307^{3}.

5741^{2} = 32959081, a zigzag square.

5741^{2} = 4059^{2} + 4060^{2}.

by Yoshio Mimura, Kobe, Japan

## 5742

5742^{2} = 32970564, a square with different digits.

3553^{k} + 4554^{k} + 5742^{k} + 6600^{k} are squares for k = 1,2,3 (143^{2}, 10483^{2}, 784927^{2}).

by Yoshio Mimura, Kobe, Japan

## 5751

5751^{2} = 126^{3} + 225^{3} + 270^{3}.

by Yoshio Mimura, Kobe, Japan

## 5752

5752^{5} = 6296430107020460032 : 6^{2} + 2^{2} + 9^{2} + 64^{2} + 3^{2} + 0^{2} + 1^{2} + 0^{2} + 7^{2} + 0^{2} + 20^{2} + 4^{2} + 6^{2} + 0^{2} + 0^{2} + 32^{2} = 5752.

by Yoshio Mimura, Kobe, Japan

## 5754

5754^{2}± 5 are primes.

5754^{2} = 205^{3} + 230^{3} + 231^{3}.

by Yoshio Mimura, Kobe, Japan

## 5756

5756^{2} = 33131536, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 5760

5760^{2} = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(5^{2} - 1)(7^{2} - 1)(9^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(9^{2} - 1)(19^{2} - 1)

= (2^{2} - 1)(5^{2} - 1)(7^{2} - 1)(9^{2} - 1)(11^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(17^{2} - 1)(31^{2} - 1)

= (5^{2} - 1)(7^{2} - 1)(9^{2} - 1)(19^{2} - 1) = (7^{2} - 1)(17^{2} - 1)(49^{2} - 1) = (11^{2} - 1)(17^{2} - 1)(31^{2} - 1).

5760^{2} = 128^{3} + 144^{3} + 304^{3} = 196^{3} + 210^{3} + 254^{3}.

5760^{2} = (55 + 56 + 57 + ... + 69)^{2} + (70 + 71 + 72 + ... + 84)^{2} + (85 + 86 + 87 + ... + 99)^{2} + ... + (175 + 176 + 177 + ... + 189)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5762

5762^{2} = 17^{4} + 41^{4} + 47^{4} + 71^{4}.

by Yoshio Mimura, Kobe, Japan

## 5765

5765^{2} = 33235225, a square consisting of just 3 kinds of digita.

5765^{2} = 33235225, 3 * 32 * 3 * 5 * 2 * 2 + 5 = 5765.

by Yoshio Mimura, Kobe, Japan

## 5766

S_{2}(5766) = S_{2}2(3921) + S_{2}(5084), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

5766^{2} = 33246756, 3 * 3 * 2 * 46 * 7 - 5 * 6 = 5766.

by Yoshio Mimura, Kobe, Japan

## 5767

5767^{2} = 33258289, 3 * 3 - 2 + 5 * 8 * 2 * 8 * 9 = 5767.

by Yoshio Mimura, Kobe, Japan

## 5768

5768^{2} = 194^{3} + 232^{3} + 238^{3}.

1 / 5768 = 0.0001733703190013, 1^{2} + 73^{2} + 3^{2} + 7^{2} + 03^{2} + 19^{2} + 001^{2} + 3^{2} = 5768.

by Yoshio Mimura, Kobe, Japan

## 5769

5769^{2} = 12^{4} + 30^{4} + 30^{4} + 75^{4}.

by Yoshio Mimura, Kobe, Japan

## 5775

5775^{2}± 2 are primes.

5775^{2} = (1^{2} + 6)(3^{2} + 6)(15^{2} + 6)(37^{2} + 6) = (27^{2} + 6)(213^{2} + 6)

= (3^{2} + 6)(7^{2} + 6)(13^{2} + 6)(15^{2} + 6).

5775^{2} = 20^{3} + 225^{3} + 280^{3}.

by Yoshio Mimura, Kobe, Japan

## 5776

The square of 76.

Page of Squares : First Upload December 25, 2007 ; Last Revised December 25, 2007by Yoshio Mimura, Kobe, Japan

## 5777

5777^{2} = 89^{3} + 166^{3} + 304^{3} = 28^{4} + 54^{4} + 54^{4} + 63^{4}.

5777 and 5993 are counter examples for the statement that every odd integer is the sum of a power of 2 and a prime (the third counter example is greater than 6*10^{5} if it exists).

by Yoshio Mimura, Kobe, Japan

## 5778

5778^{2} = (2^{2} + 2)(31^{2} + 2)(76^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 5779

5779^{2} = 33396841, 3^{2} + 3^{2} + 3^{2} + 9^{2} + 6^{2} + 8^{2} + 4^{2} + 1^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 5780

5780^{2} = (8^{2} + 4)(9^{2} + 4)(76^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 5781

5781^{2} = 2^{3} + 145^{3} + 312^{3}.

5781^{4} = 1116893793241521,

and 1^{2} + 11^{2} + 68^{2} + 9^{2} + 3^{2} + 7^{2} + 9^{2} + 3^{2} + 24^{2} + 15^{2} + 2^{2} + 1^{2} = 5781,

5781^{4} = 1116893793241521,

and 11^{2} + 1^{2} + 68^{2} + 9^{2} + 3^{2} + 7^{2} + 9^{2} + 3^{2} + 24^{2} + 15^{2} + 2^{2} + 1^{2} = 5781.

by Yoshio Mimura, Kobe, Japan

## 5782

5782^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 5783

5783^{2} = 54^{3} + 170^{3} + 305^{3}.

by Yoshio Mimura, Kobe, Japan

## 5785

5785^{2} = (29^{2} + 4)(199^{2} + 4).

Loop of length 35 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

5785 - 10474 - 5493 - 11565 - ... - 7034 - 6056 - 6736 - 5785

(Note f(5785) = 57^{2} + 85^{2} = 10474, f(10474) = 1^{2} + 04^{2} + 74^{2} = 5493, etc. See 37)

5785^{2} = 33466225 appears in the decimal expressions of e:

e = 2.71828•••33466225••• (from the 20869th digit)

(33466225 is the second 8-digit square in the expression of e.)

by Yoshio Mimura, Kobe, Japan

## 5790

5790^{2} = 109^{3} + 164^{3} + 303^{3}.

by Yoshio Mimura, Kobe, Japan

## 5795

5795 = (1^{2} + 2^{2} + 3^{2} + ... + 152^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 8^{2}).

5795^{2} = 33582025, 3 - 3 + 5820 - 25 = 3 / 3 * 5820 - 25 = 5795.

by Yoshio Mimura, Kobe, Japan

## 5796

5796 = (1^{2} + 2^{2} + 3^{2} + ... + 80^{2}) / (1^{2} + 2^{2} + 3^{2} + 4^{2}).

5796^{2} = 130^{3} + 140^{3} + 306^{3}.

The integral triangle of sides 1820, 83441, 85077 has square area 5796^{2}.

by Yoshio Mimura, Kobe, Japan

## 5797

5797 = (1^{2} + 2^{2} + 3^{2} + ... + 170^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 9^{2}).

5797^{2} = 81^{3} + 220^{3} + 282^{3} = 165^{3} + 219^{3} + 265^{3}.

by Yoshio Mimura, Kobe, Japan