## 3700

3700^{2} = (1^{2} + 1)(3^{2} + 1)(7^{2} + 1)(117^{2} + 1) = (1^{2} + 4)(12^{2} + 4)(136^{2} + 4).

3700^{2} + 3701^{2} + 3702^{2} + ... + 5384^{2} = 5385^{2} + 5386^{2} + 5387^{2} + ... + 6395^{2}.

by Yoshio Mimura, Kobe, Japan

## 3701

3701^{2} = 13697401, 1 + 3697 + 4 + 0 - 1 = 1 * 3697 + 4 + 0 * 1 = 3701.

by Yoshio Mimura, Kobe, Japan

## 3702

3702^{5} = 695315758215496032 : 6^{2} + 9^{2} + 53^{2} + 1^{2} + 5^{2} + 7^{2} + 5^{2} + 8^{2} + 21^{2} + 5^{2} + 4^{2} + 9^{2} + 6^{2} + 0^{2} + 3^{2} + 2^{2} = 3702.

by Yoshio Mimura, Kobe, Japan

## 3703

3703^{2} = 23^{4} + 46^{4} + 46^{4} + 46^{4} = 34^{4} + 44^{4} + 44^{4} + 47^{4}.

3703^{2} = S_{2}(213) + S_{2}(315), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

3703^{2} = 13712209, 1 * 3712 + 2 * 0 - 9 = 1 * 3712 - 2 * 0 - 9 = 3703.

by Yoshio Mimura, Kobe, Japan

## 3704

3704^{2} = 42^{3} + 134^{3} + 224^{3} = 8^{5} + 8^{5} + 14^{5} + 22^{5} + 24^{5}.

by Yoshio Mimura, Kobe, Japan

## 3708

3708^{2} = 12^{4} + 42^{4} + 48^{4} + 48^{4}.

3708^{2} = 13749264, 1 - 37 + 4 * 9 * 26 * 4 = 3708.

by Yoshio Mimura, Kobe, Japan

## 3709

3709^{2} = 13756681, 1 + 3756 - 6 * 8 * 1 = 1 * 3756 - 6 * 8 + 1 = 3709.

by Yoshio Mimura, Kobe, Japan

## 3710

3710^{2} = (1^{2} + 6)(2^{2} + 6)(10^{2} + 6)(43^{2} + 6) = (8^{2} + 6)(10^{2} + 6)(43^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 3711

3711^{2} = 71^{3} + 159^{3} + 211^{3}.

3711^{2} = 13771521, 13 + 7 + 71 * 52 - 1 = 3711.

by Yoshio Mimura, Kobe, Japan

## 3712

3712^{2} = 125^{3} + 164^{3} + 195^{3}.

3712^{2} = (5^{2} + 7)(15^{2} + 7)(43^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 3714

3714^{2} = 13793796, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 3716

3716^{2} = 40^{3} + 158^{3} + 214^{3}.

by Yoshio Mimura, Kobe, Japan

## 3717

3717^{2} = 13816089, 1^{2} + 3^{2} + 8^{2} + 1^{2} + 6^{2} + 0^{2} + 8^{2} + 9^{2} = 16^{2}.

3717^{2} = (65 + 66 + 67)^{2} + (68 + 69 + 70)^{2} + (71 + 72 + 73)^{2} + ... + (239 + 240 + 241)^{2}.

by Yoshio Mimura, Kobe, Japan

## 3718

3718^{2} = 55^{3} + 142^{3} + 221^{3}.

1 / 3718 = 0.00026896..., and 26896 = 164^{2}.

by Yoshio Mimura, Kobe, Japan

## 3721

The square of 61.

3721^{2} = 13845841, 1^{2} + 3^{2} + 8^{2} + 4^{2} + 5^{2} + 8^{2} + 4^{2} + 1^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 3722

3722^{2} = 227^{3} + 9^{5} + 8^{7}.

by Yoshio Mimura, Kobe, Japan

## 3723

3723^{2} = 13860729, a square with different digits.

3723^{2}± 2 are primes.

3723^{2} = 30^{3} + 188^{3} + 193^{3} = 40^{3} + 134^{3} + 225^{3}.

by Yoshio Mimura, Kobe, Japan

## 3724

3724^{2} = (2^{2} + 3)(23^{2} + 3)(61^{2} + 3) = (2^{2} + 3)(4^{2} + 3)(5^{2} + 3)(61^{2} + 3) = (4^{2} + 3)(23^{2} + 3)(37^{2} + 3).

3724^{2} = S_{2}(14) + S_{2}(346), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 3725

3725^{2} = 13875625, 1 * 3875 - 6 * 25 = 3725.

Komachi equation: 3725^{2} = 9^{2} / 87^{2} / 6^{2} * 5^{2} * 43210^{2}.

by Yoshio Mimura, Kobe, Japan

## 3726

3726^{2} = 34^{3} + 129^{3} + 227^{3}.

3726^{2} = (1^{2} + 5)(2^{2} + 5)(8^{2} + 5)(61^{2} + 5) = (2^{2} + 5)(7^{2} + 5)(169^{2} + 5) = (7^{2} + 5)(8^{2} + 5)(61^{2} + 5).

3726^{2} = 178^{2} + 179^{2} + 180^{2} + 181^{2} + ... + 361^{2}.

by Yoshio Mimura, Kobe, Japan

## 3728

3728^{2} = 13897984, 1 + 38 * 97 + 9 + 8 * 4 = 3728.

by Yoshio Mimura, Kobe, Japan

## 3729

The square root of 3729 is 61.065..., 61 = 0^{2} + 6^{2} + 5^{2}.

by Yoshio Mimura, Kobe, Japan

## 3732

3732^{2} = 1^{3} + 47^{3} + 240^{3} = 66^{3} + 128^{3} + 226^{3}.

by Yoshio Mimura, Kobe, Japan

## 3734

3734^{2} = 13942756, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 3738

3738^{2} = (1^{2} + 5)(23^{2} + 5)(66^{2} + 5) = (1^{2} + 5)(4^{2} + 5)(333^{2} + 5) = (11^{2} + 5)(333^{2} + 5)

= (2^{2} + 5)(3^{2} + 5)(333^{2} + 5).

3738^{2} = 113^{3} + 171^{3} + 196^{3}.

by Yoshio Mimura, Kobe, Japan

## 3740

3740^{2} = S_{2}(4) + S_{2}(347), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

3740^{5} = 731742047062400000 : 7^{2} + 31^{2} + 7^{2} + 4^{2} + 20^{2} + 47^{2} + 0^{2} + 6^{2} + 2^{2} + 4^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 3740.

by Yoshio Mimura, Kobe, Japan

## 3741

3741^{2} + 3742^{2} + 3743^{2} + ... + 3784^{2} = 3785^{2} + 3786^{2} + 3787^{2} + ... + 3827^{2}.

3741^{5} = 732720835107334701 : 7^{2} + 3^{2} + 27^{2} + 20^{2} + 8^{2} + 35^{2} + 1^{2} + 0^{2} + 7^{2} + 3^{2} + 34^{2} + 7^{2} + 0^{2} + 1^{2} = 3741.

by Yoshio Mimura, Kobe, Japan

## 3743

3743^{2} = 8^{4} + 21^{4} + 48^{4} + 54^{4}.

by Yoshio Mimura, Kobe, Japan

## 3744

3744^{2} = (3^{2} - 1)(25^{2} - 1)(53^{2} - 1) = (1^{3} + 8)(2^{3} + 8)(46^{3} + 8).

by Yoshio Mimura, Kobe, Japan

## 3746

3746^{2} = 14032516, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 3747

3747^{2} = 68^{3} + 97^{3} + 234^{3}.

by Yoshio Mimura, Kobe, Japan

## 3748

3748^{2} = 116^{3} + 179^{3} + 189^{3}.

by Yoshio Mimura, Kobe, Japan

## 3749

3749^{2} = S_{2}(207) + S_{2}(321), S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 3750

3750^{2} = (4^{2} + 9)(9^{2} + 9)(79^{2} + 9).

3750, 3751, 3752 and 3753 are four consecutive integers having square factors (the third case).

Page of Squares : First Upload November 10, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan

## 3751

The square root of 3751 is 61.2454..., and 61 = 2^{2} + 4^{2} + 5^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 3753

3753^{2} = 8^{4} + 8^{4} + 51^{4} + 52^{4} = 24^{4} + 36^{4} + 48^{4} + 51^{4}.

by Yoshio Mimura, Kobe, Japan

## 3754

3754^{2} = 14092516, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 3755

11^{2} + 24^{2} + 37^{2} + ... + (13x+11)^{2} + ... + 3742^{2} + 3755^{2} = 36941^{2}.

by Yoshio Mimura, Kobe, Japan

## 3756

3756^{2} = 14107536, 1 + 4 - 1 + 0 + 7 * 536 = 1 * 4 + 1 * 0 + 7 * 536 = 1 * 4 - 1 * 0 + 7 * 536

= 1 * 4 * 1 + 0 + 7 * 536 = 14 - 10 + 7 * 536 = 3756.

by Yoshio Mimura, Kobe, Japan

## 3757

3757^{2} = (2^{2} + 9)(1042^{2} + 9).

3757^{4} = 199234608272401, and 1^{2} + 9^{2} + 9^{2} + 2^{2} + 34^{2} + 6^{2} + 0^{2} + 8^{2} + 27^{2} + 2^{2} + 40^{2} + 1^{2} =3757 .

by Yoshio Mimura, Kobe, Japan

## 3758

3758^{3} = 53072595512, and 53^{2} + 0^{2} + 7^{2} + 25^{2} + 9^{2} + 5^{2} + 5^{2} + 12^{2} = 3758.

by Yoshio Mimura, Kobe, Japan

## 3762

3762^{2} = (1^{2} + 2)(2^{2} + 2)(14^{2} + 2)(63^{2} + 2) = (1^{2} + 2)(2^{2} + 2)(3^{2} + 2)(4^{2} + 2)(63^{2} + 2)

= (1^{2} + 2)(3^{2} + 2)(4^{2} + 2)(6^{2} + 2)(25^{2} + 2) = (1^{2} + 2)(3^{2} + 2)(6^{2} + 2)(8^{2} + 2)(13^{2} + 2)

= (1^{2} + 2)(4^{2} + 2)(8^{2} + 2)(63^{2} + 2) = (1^{2} + 2)(6^{2} + 2)(14^{2} + 2)(25^{2} + 2)

= (2^{2} + 2)(6^{2} + 2)(13^{2} + 2)(19^{2} + 2) = (3^{2} + 2)(6^{2} + 2)(13^{2} + 2)(14^{2} + 2)

= (3^{2} + 2)(6^{2} + 2)(184^{2} + 2) = (4^{2} + 2)(14^{2} + 2)(63^{2} + 2) = (6^{2} + 2)(19^{2} + 2)(32^{2} + 2).

The integral triangle of sides 369, 263177, 263530 has square area 3762^{2}.

by Yoshio Mimura, Kobe, Japan

## 3764

3764^{2} = 14167696, 1^{2} + 4^{2} + 1^{2} + 6^{2} + 7^{2} + 6^{2} + 9^{2} + 6^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 3765

3765^{2} = 14175225, 14 + 1 + 75 * 2 * 25 = 3765.

by Yoshio Mimura, Kobe, Japan

## 3766

3766^{2} = 14182756, a zigzag square.

3766^{2} = 14182756, 1^{2} + 4^{2} + 1^{2} + 8^{2} + 2^{2} + 7^{2} + 5^{2} + 6^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 3768

3768^{2} = 14197824, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 3770

3770^{2} = (1^{2} + 1)(2^{2} + 1)(12^{2} + 1)(99^{2} + 1) = (1^{2} + 1)(2^{2} + 1)(17^{2} + 1)(70^{2} + 1)

= (1^{2} + 1)(3^{2} + 1)(12^{2} + 1)(70^{2} + 1) = (2^{2} + 1)(17^{2} + 1)(99^{2} + 1)

= (2^{2} + 1)(5^{2} + 1)(8^{2} + 1)(41^{2} + 1) = (3^{2} + 1)(12^{2} + 1)(99^{2} + 1)

= (3^{2} + 1)(17^{2} + 1)(70^{2} + 1) = (5^{2} + 1)(18^{2} + 1)(41^{2} + 1) = (5^{2} + 4)(24^{2} + 4)(29^{2} + 4).

3770^{2} = 38^{2} + 39^{2} + 40^{2} + 41^{2} + 42^{2} + 43^{2} + ... + 349^{2}.

Loop of length 5 (See 41):

3770 -- 37^{2} + 70^{2} = 6269 -- 62^{2} + 69^{2} = 8605 -- 86^{2} + 05^{2} = 7421 -- 74^{2} + 21^{2} = 5917 - 59^{2} + 17^{2} = 3770

by Yoshio Mimura, Kobe, Japan

## 3772

3772^{4} = 202435528704256, and 2^{2} + 0^{2} + 24^{2} + 35^{2} + 5^{2} + 2^{2} + 8^{2} + 7^{2} + 0^{2} + 42^{2} + 5^{2} + 6^{2} = 3772.

by Yoshio Mimura, Kobe, Japan

## 3773

3773^{2} = (123 + 124 + 125 + 126 + 127 + 128 + 129)^{2} + (130 + 131 + 132 + 133 + 134 + 135 + 136)^{2} + (137 + 138 + 139 + 140 + 141 + 142 + 143)^{2} + ... + (193 + 194 + 195 + 196 + 197 + 198 + 199)^{2}.

by Yoshio Mimura, Kobe, Japan

## 3775

3775^{2} = 14250625, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 3776

3776^{2} = 32^{3} + 144^{3} + 224^{3}.

3776^{2} = 14258176, 1^{2} + 4^{2} + 2^{2} + 5^{2} + 8^{2} + 1^{2} + 7^{2} + 6^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 3777

3777^{2} = 14265729, a zigzag square.

3777^{2} = 7^{4} + 46^{4} + 46^{4} + 48^{4}.

by Yoshio Mimura, Kobe, Japan

## 3779

3779^{2} = 64^{3} + 138^{3} + 225^{3}.

by Yoshio Mimura, Kobe, Japan

## 3780

3780^{2} = (1^{2} + 5)(2^{2} + 5)(3^{2} + 5)(5^{2} + 5)(25^{2} + 5) = (1^{2} + 5)(3^{2} + 5)(5^{2} + 5)(7^{2} + 5)(10^{2} + 5)

= (1^{2} + 5)(5^{2} + 5)(11^{2} + 5)(25^{2} + 5) = (2^{2} - 1)(4^{2} - 1)(8^{2} - 1)(71^{2} - 1) = (2^{2} - 1)(9^{2} - 1)(244^{2} - 1)

= (3^{2} + 5)(5^{2} + 5)(7^{2} + 5)(25^{2} + 5).

Komachi equations:

3780^{2} = 12^{2} * 3^{2} / 4^{2} * 5^{2} / 6^{2} * 7^{2} * 8^{2} * 9^{2} = 12^{2} / 3^{2} * 4^{2} * 5^{2} * 6^{2} * 7^{2} / 8^{2} * 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 3783

3783^{2} = 2^{4} + 22^{4} + 51^{4} + 52^{4} = 13^{4} + 34^{4} + 42^{4} + 56^{4}.

The square root of 3783 is 61.506..., and 61 = 5^{2} + 0^{2} + 6^{2}.

by Yoshio Mimura, Kobe, Japan

## 3784

3784^{2} = 14318656, 1 * 431 * 8 + 6 * 56 = 3784.

by Yoshio Mimura, Kobe, Japan

## 3786

3786^{2} = 102^{3} + 167^{3} + 205^{3}.

by Yoshio Mimura, Kobe, Japan

## 3787

3787^{2} = 14341369, 1^{2} + 4^{2} + 3^{2} + 4^{2} + 1^{2} + 3^{2} + 6^{2} + 9^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 3788

3788^{2} = 3^{0} + 3^{2} + 3^{3} + 3^{15}.

by Yoshio Mimura, Kobe, Japan

## 3789

3789^{2} = 61^{3} + 171^{3} + 209^{3}.

by Yoshio Mimura, Kobe, Japan

## 3790

The square root of 3790 is 61.56..., and 61 = 5^{2} + 6^{2}.

by Yoshio Mimura, Kobe, Japan

## 3792

3792^{2} = 14379264.

3792^{2} = 14379264, 1 - 4 + 3 + 79 * 2 * 6 * 4 = 3792.

by Yoshio Mimura, Kobe, Japan

## 3793

3793^{2} = 14386849, a zigzag square.

A cubic polynomial :

(X + 1804^{2})(X + 2112^{2})(X + 2583^{2}) = X^{3} + 3793^{2}X^{2} + 8123412^{2}X + 9841353984^{2}.

1 / 3793 = 0.000263643553915, 26^{2} + 36^{2} + 4^{2} + 3^{2} + 5^{2} + 5^{2} + 39^{2} + 15^{2} = 3793.

by Yoshio Mimura, Kobe, Japan

## 3795

3795^{2} = 66^{3} + 137^{3} + 226^{3} = 132^{3} + 170^{3} + 193^{3}.

3795^{2} = 14402025, 144 = 12^{2} and 2025 = 45^{2}.

3795^{2} = 512^{2} + 513^{2} + 514^{2} + 515^{2} + 516^{2} + 517^{2} + ... + 561^{2}.

3795^{2} = (6^{3} + 9)(40^{3} + 9).

by Yoshio Mimura, Kobe, Japan

## 3796

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

3796 - 10585 - 7251 - 7785 - ... - 2570 - 5525 - 3650 - 3796

(Note f(3796) = 37^{2} + 96^{2} = 10585, f(10585) = 1^{2} + 05^{2} + 85^{2} = 7251, etc. See 41)

by Yoshio Mimura, Kobe, Japan