## 4702

4702^{2} = 1^{2} + 17^{2} + 37^{2} + 67^{2}

by Yoshio Mimura, Kobe, Japan

## 4703

4703^{2} = 2^{2} + 23^{2} + 26^{2} + 68^{2}

by Yoshio Mimura, Kobe, Japan

## 4704

4704^{2} = 56^{2} + 280^{2}

A cubic polynomial :

(X + 65^{2})(X + 72^{2})(X + 4704^{2}) = X^{3} + 4705^{2}X^{2} + 456312^{2}X + 22014720^{2}.

70^{2} + 4704 = 98^{2}, 70^{2} - 4704 = 14^{2}.

Komachi equations:

4704^{2} = 1^{2} * 2^{2} / 3^{2} / 4^{2} * 56^{2} * 7^{2} * 8^{2} * 9^{2} = 9^{2} * 8^{2} * 7^{2} / 6^{2} / 5^{2} * 4^{2} / 3^{2} * 210^{2}

= 98^{2} / 7^{2} * 6^{2} / 5^{2} * 4^{2} / 3^{2} * 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 4705

A cubic polynomials :

(X + 65^{2})(X + 72^{2})(X + 4704^{2}) = X^{3} + 4705^{2}X^{2} + 456312^{2}X + 22014720^{2},

(X + 560^{2})(X + 1863^{2})(X + 4284^{2}) = X^{3} + 4705^{2}X^{2} + 8398908^{2}X + 4469411520^{2}.

by Yoshio Mimura, Kobe, Japan

## 4707

4707^{2} = 44^{2} + 217^{2} + 228^{2}

by Yoshio Mimura, Kobe, Japan

## 4709

4709^{2} = 22174681, 2 * 2 * 1 * 7 + 4681 = 22 - 1 + 7 + 4681 = 4709.

4709^{4} = 491716477451761,

and 4^{2} + 9^{2} + 1^{2} + 7^{2} + 1^{2} + 64^{2} + 7^{2} + 7^{2} + 4^{2} + 5^{2} + 17^{2} + 6^{2} + 1^{2} = 4709,

4709^{4} = 491716477451761,

and 4^{2} + 9^{2} + 17^{2} + 1^{2} + 64^{2} + 7^{2} + 7^{2} + 4^{2} + 5^{2} + 1^{2} + 7^{2} + 6^{2} + 1^{2} = 4709.

by Yoshio Mimura, Kobe, Japan

## 4710

4710^{2} = 78^{2} + 97^{2} + 275^{2}

by Yoshio Mimura, Kobe, Japan

## 4712

4712^{2} = 60^{2} + 164^{2} + 260^{2} = 38^{2} + 42^{2} + 54^{2} + 54^{2}

The square root of 4712 is 68.644..., and 68 = 6^{2} + 4^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 4715

4715^{2} = 116^{2} + 193^{2} + 238^{2}

4715^{2} = (122 + 123 + 124 + ... + 144)^{2} + (145 + 146 + 147 + ... + 167)^{2} + (168 + 169 + 170 + ... + 190)^{2} + ... + (145 + 146 + 147 + ... + 167)2.

by Yoshio Mimura, Kobe, Japan

## 4721

4721^{2} = 25^{2} + 40^{2} + 40^{2} + 64^{2}.

by Yoshio Mimura, Kobe, Japan

## 4724

4724^{2} = 30^{2} + 198^{2} + 244^{2} = 50^{2} + 220^{2} + 226^{2}

1 / 4724 = 0.0002116... and 2116 = 46^{2}. (also 4725).

The square root of 4724 is 68.7313..., and 68 = 7^{2} + 3^{2} + 1^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 4725

4725^{2} = (4^{2} - 1)(6^{2} - 1)(8^{2} - 1)(26^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 4728

4728^{2} = 77^{2} + 142^{2} + 267^{2}

by Yoshio Mimura, Kobe, Japan

## 4732

4732^{2} = (1^{2} + 3)(19^{2} + 3)(124^{2} + 3) = (1^{2} + 3)(2^{2} + 3)(7^{2} + 3)(124^{2} + 3)

= (5^{2} + 3)(7^{2} + 3)(124^{2} + 3).

4732^{2} = 26^{2} + 52^{2} + 52^{2} + 52^{2}

by Yoshio Mimura, Kobe, Japan

## 4733

4733^{2} = 105^{2} + 198^{2} + 238^{2}

by Yoshio Mimura, Kobe, Japan

## 4734

1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + ... + 4734^{2} = 35375379135, the first 11-digit sum whose digits are odd (there are 3 such sums in all. See 5382, 6102).

by Yoshio Mimura, Kobe, Japan

## 4735

4735^{2} = 140^{2} + 193^{2} + 232^{2}

by Yoshio Mimura, Kobe, Japan

## 4737

The square root of 4737 is 68.82..., and 68 = 8^{2} + 2^{2}.

by Yoshio Mimura, Kobe, Japan

## 4738

4738^{2} = 22448644, a square with even digits.

by Yoshio Mimura, Kobe, Japan

## 4742

4742^{2}± 3 are primes.

4742^{2} = 7^{2} + 141^{2} + 270^{2}.

by Yoshio Mimura, Kobe, Japan

## 4743

4743^{2} = 6^{2} + 41^{2} + 56^{2} + 56^{2}

4743^{2} = 22496049, and 22 * 4 * 9 * 6 + 0 * 4 - 9 = 4743.

by Yoshio Mimura, Kobe, Japan

## 4744

4744^{2} = 40^{2} + 174^{2} + 258^{2}.

by Yoshio Mimura, Kobe, Japan

## 4745

4745^{2} = (2^{2} + 9)(8^{2} + 9)(154^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 4746

4746^{2} = 9^{2} + 83^{2} + 280^{2}.

by Yoshio Mimura, Kobe, Japan

## 4747

1 / 4747 = 0.000210659, 21^{2} + 0^{2} + 65^{2} + 9^{2} = 4747,

1 / 4747 = 0.000210659, 21^{2} + 065^{2} + 9^{2} = 4747.

by Yoshio Mimura, Kobe, Japan

## 4750

4750^{2} = 22562500, and 225 = 15^{2}, 62500 = 250^{2}.

by Yoshio Mimura, Kobe, Japan

## 4752

4752^{2}± 5 are primes.

4752^{2} = (1^{2} + 8)(2^{2} + 8)(16^{2} + 8)(28^{2} + 8) = (1^{2} + 8)(2^{2} + 8)(4^{2} + 8)(5^{2} + 8)(16^{2} + 8)

= (1^{2} + 8)(4^{2} + 8)(5^{2} + 8)(6^{2} + 8)(8^{2} + 8) = (1^{2} + 8)(4^{2} + 8)(8^{2} + 8)(38^{2} + 8)

= (1^{2} + 8)(6^{2} + 8)(8^{2} + 8)(28^{2} + 8) = (10^{2} + 8)(16^{2} + 8)(28^{2} + 8)

= (2^{2} + 8)(4^{2} + 8)(16^{2} + 8)(17^{2} + 8) = (2^{2} + 8)(4^{2} + 8)(280^{2} + 8) = (2^{2} + 8)(5^{2} + 8)(8^{2} + 8)(28^{2} + 8) = (4^{2} + 8)(5^{2} + 8)(10^{2} + 8)(16^{2} + 8) = (4^{2} + 8)(6^{2} + 8)(8^{2} + 8)(17^{2} + 8) = (6^{2} + 8)(16^{2} + 8)(44^{2} + 8).

4752^{2} = 19^{2} + 53^{2} + 282^{2} = 108^{2} + 186^{2} + 246^{2} = 111^{2} + 169^{2} + 254^{2}

4752^{2} = 22581504 appears in the decimal expressions of e:

e = 2.71828•••22581504••• (from the 29097th digit)

(22581504 is the fifth 8-digit square in the expression of e.)

by Yoshio Mimura, Kobe, Japan

## 4753

4753^{2} = 49^{2} + 154^{2} + 266^{2} = 1^{2} + 34^{2} + 46^{2} + 64^{2} = 28^{2} + 42^{2} + 42^{2} + 63^{2}

4753^{2} = 22591009, and 2^{2} + 2^{2} + 5^{2} + 9^{2} + 1^{2} + 0^{2} + 0^{2} + 9^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 4756

4756^{2}± 3 are primes.

4756^{2} = 22619536, and 2^{2} + 2^{2} + 6^{2} + 1^{2} + 9^{2} + 5^{2} + 3^{2} + 6^{2} = 14^{2}.

Loop of length 4 :

4756 --- 47^{2} + 56^{2} = 5345 --- 53^{2} + 45^{2} = 4834 --- 48^{2} + 34^{2} = 3460 --- 34^{2} + 60^{2} = 4756

by Yoshio Mimura, Kobe, Japan

## 4759

4759^{2} = 20^{2} + 20^{2} + 46^{2} + 65^{2}

by Yoshio Mimura, Kobe, Japan

## 4760

4760^{2}= 431 x 432 + 432 x 433 + 433 x 434 + 434 x 435 + ... + 528 x 529.

Komachi equation: 4760^{2} = 98^{2} * 765^{2} * 4^{2} / 3^{2} / 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 4761

The square of 69.

4761^{2} = 84^{2} + 120^{2} + 273^{2}

by Yoshio Mimura, Kobe, Japan

## 4762

The square root of 4762 is 69.00724..., and 69 = 0^{2} + 0^{2} + 7^{2} + 2^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 4763

4763^{5} = 2451336362939320043 : 2^{2} + 45^{2} + 13^{2} + 3^{2} + 6^{2} + 36^{2} + 2^{2} + 9^{2} + 3^{2} + 9^{2} + 32^{2} + 0^{2} + 0^{2} + 4^{2} + 3^{2} = 2^{2} + 45^{2} + 13^{2} + 36^{2} + 3^{2} + 6^{2} + 2^{2} + 9^{2} + 3^{2} + 9^{2} + 32^{2} + 0^{2} + 0^{2} + 4^{2} + 3^{2} = 4763.

by Yoshio Mimura, Kobe, Japan

## 4765

A cubic polynomial :

(X + 32^{2})(X + 2205^{2})(X + 4224^{2}) = X^{3} + 4765^{2}X^{2} + 9315168^{2}X + 298045440^{2}.

by Yoshio Mimura, Kobe, Japan

## 4766

4766^{2} = 22714756, and 2 + 2 + 7 - 1 + 4756 = 2 * 2 + 7 - 1 + 4756 = 4766.

by Yoshio Mimura, Kobe, Japan

## 4767

4767^{2} = 2^{2} + 9^{2} + 34^{2} + 68^{2}

4767^{2} = (150 + 151 + 152)^{2} + (153 + 154 + 155)^{2} + ... + (294 + 295 + 296)^{2}.

by Yoshio Mimura, Kobe, Japan

## 4768

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

4768 - 6833 - 5713 - 3418 - ... - 2838 - 2228 - 1268 - 4768

(Note f(4768) = 47^{2} + 68^{2} = 6833, f(6833) = 68^{2} + 33^{2} = 5713, etc. See 1268)

by Yoshio Mimura, Kobe, Japan

## 4770

4770^{2} = (3^{2} + 9)(16^{2} + 9)(69^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 4771

The square root of 4771 is 69.0724..., and 69 = 0^{2} + 7^{2} + 2^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 4772

4772^{5} = 2474583854273733632 : 2^{2} + 4^{2} + 7^{2} + 45^{2} + 8^{2} + 3^{2} + 8^{2} + 5^{2} + 4^{2} + 2^{2} + 7^{2} + 37^{2} + 3^{2} + 3^{2} + 6^{2} + 32^{2} = 2^{2} + 4^{2} + 7^{2} + 4^{2} + 5^{2} + 8^{2} + 3^{2} + 8^{2} + 5^{2} + 42^{2} + 7^{2} + 37^{2} + 3^{2} + 36^{2} + 3^{2} + 2^{2} = 4772.

by Yoshio Mimura, Kobe, Japan

## 4775

4775^{2} = 22800625, 2 - 2 + 800 * 6 - 25 = 2 / 2 * 800 * 6 - 25 = 4775.

by Yoshio Mimura, Kobe, Japan

## 4776

4776^{2} = 68^{2} + 160^{2} + 264^{2}

by Yoshio Mimura, Kobe, Japan

## 4779

4779^{2} = (113 + 114 + 115)^{2} + (116 + 117 + 118)^{2} + (119 + 120 + 121)^{2} + ... + (287 + 288 + 289)^{2}.

by Yoshio Mimura, Kobe, Japan

## 4780

4780^{2} = 10^{2} + 218^{2} + 232^{2}

by Yoshio Mimura, Kobe, Japan

## 4782

4782^{5} = 2500620912904734432 : 2^{2} + 5^{2} + 0^{2} + 0^{2} + 6^{2} + 20^{2} + 9^{2} + 1^{2} + 29^{2} + 0^{2} + 47^{2} + 34^{2} + 4^{2} + 3^{2} + 2^{2} = 4782.

by Yoshio Mimura, Kobe, Japan

## 4783

4783^{2} = 2^{2} + 18^{2} + 18^{2} + 69^{2}

by Yoshio Mimura, Kobe, Japan

## 4784

4784^{5} = 2505854525373415424 : 2^{2} + 5^{2} + 0^{2} + 5^{2} + 8^{2} + 5^{2} + 45^{2} + 2^{2} + 5^{2} + 37^{2} + 34^{2} + 1^{2} + 5^{2} + 4^{2} + 2^{2} + 4^{2} = 4784.

by Yoshio Mimura, Kobe, Japan

## 4785

4785^{2} = (9^{2} + 6)(513^{2} + 6).

4785^{4} = 524237119250625, and 5^{2} + 2^{2} + 4^{2} + 2^{2} + 37^{2} + 11^{2} + 9^{2} + 2^{2} + 50^{2} + 6^{2} + 25^{2} = 4785.

4785^{k} + 12265^{k} + 14355^{k} + 16995^{k} are squares for k = 1,2,3 (220^{2}, 25850^{2}, 3133900^{2}).

by Yoshio Mimura, Kobe, Japan

## 4788

4788^{2} = 72^{2} + 173^{2} + 259^{2}

4788^{2} = 22924944, a square consisting of just 3 kinds of digits.

Komachi equation: 4788^{2} = 9^{2} / 8^{2} * 76^{2} / 5^{2} * 4^{2} / 3^{2} * 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 4789

4789^{2} = 58^{2} + 82^{2} + 281^{2}

4789^{2} = 22934521, and 2^{2} + 2^{2} + 9^{2} + 3^{2} + 4^{2} + 5^{2} + 2^{2} + 1^{2} = 12^{2}.

by Yoshio Mimura, Kobe, Japan

## 4795

19^{2} + 43^{2} + 67^{2} + 91^{2} + 115^{2} + ... + (24n + 19)^{2} + ... + 1171^{2} = 4795^{2}.

by Yoshio Mimura, Kobe, Japan

## 4796

4796^{2}± 3 are primes.

4796^{2} = 27^{2} + 120^{2} + 277^{2}

S_{2}(4796) = S_{2}(3531) + S_{2}(4047), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

4796^{2} = 23001616, and 2 + 300 * 16 - 1 * 6 = 2 + 300 * 16 * 1 - 6 = 4796.

by Yoshio Mimura, Kobe, Japan

## 4797

4797^{2} = 23011209, and 2^{2} + 3^{2} + 0^{2} + 1^{2} + 1^{2} + 2^{2} + 0^{2} + 9^{2} = 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 4798

4798^{2} = 23020804, a zigzag square.

4798^{2} = 23020804, and 2 + 30 * 20 * 8 + 0 - 4 = 4798.

by Yoshio Mimura, Kobe, Japan

## 4799

4799^{2} = 23030401, a zigzag square.

The square root of 4799 is 69.274..., and 69 = 2^{2} + 7^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan