## 3400

3400^{2} = (1^{2} + 4)(6^{2} + 4)(9^{2} + 4)(26^{2} + 4) = (9^{2} + 4)(14^{2} + 4)(26^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 3401

3401^{2} = 20^{3} + 25^{3} + 226^{3}.

by Yoshio Mimura, Kobe, Japan

## 3402

3402^{2} = (1^{2} + 5)(2^{2} + 5)(4^{2} + 5)(101^{2} + 5) = (2^{2} + 5)(11^{2} + 5)(101^{2} + 5) = (4^{2} + 5)(7^{2} + 5)(101^{2} + 5).

3402^{2} = 29^{3} + 159^{3} + 196^{3}.

Komachi equation: 3402^{2} = 9^{2} * 87^{2} * 6^{2} - 54^{2} * 3^{2} * 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 3403

3403 = (1^{2} + 2^{2} + 3^{2} + ... + 82^{2}) / (1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2}).

3403^{2} = 148^{3} + 152^{3} + 169^{3}.

3403^{2} + 3404^{2} + 3405^{2} + ... + 3444^{2} = 3445^{2} + 3446^{2} + 3447^{2} + ... + 3485^{2}.

by Yoshio Mimura, Kobe, Japan

## 3405

3405^{2} = 86^{3} + 105^{3} + 214^{3} = 1^{5} + 14^{5} + 17^{5} + 20^{5} + 23^{5}.

by Yoshio Mimura, Kobe, Japan

## 3406

3406^{2} = 15^{2} + 16^{2} + 17^{2} + 18^{2} + 19^{2} + 20^{2} + ... + 326^{2}.

by Yoshio Mimura, Kobe, Japan

## 3407

3407^{2} = 20^{3} + 174^{3} + 185^{3}.

1 / 3407 = 0.00029351335485, 29^{2} + 3^{2} + 5^{2} + 13^{2} + 3^{2} + 5^{2} + 48^{2} + 5^{2} = 3407.

by Yoshio Mimura, Kobe, Japan

## 3408

3408^{2} = 11614464, a square with just 3 kinds of digits.

3408^{2} = 120^{3} + 148^{3} + 188^{3}.

The square root of 3408 is 58.37...., 58 = 3^{2} + 7^{2}.

by Yoshio Mimura, Kobe, Japan

## 3409

3409^{2} = 11621281, 1 + 1 * 6 + 21 * 2 * 81 = 1 * 1 + 6 + 21 * 2 * 81 = 3409.

by Yoshio Mimura, Kobe, Japan

## 3410

3410^{2} = (2^{2} + 6)(5^{2} + 6)(7^{2} + 6)(26^{2} + 6) = (4^{2} + 6)(7^{2} + 6)(98^{2} + 6).

3410^{2} = 83^{3} + 169^{3} + 184^{3}.

by Yoshio Mimura, Kobe, Japan

## 3411

3411^{2} = 89^{3} + 171^{3} + 181^{3} = 90^{3} + 166^{3} + 185^{3} = 111^{3} + 129^{3} + 201^{3}.

by Yoshio Mimura, Kobe, Japan

## 3414

3414^{2} = 11655396, 1 - 1 + 6 * 553 + 96 = 1 * 1 * 6 * 553 + 96 = 3414.

by Yoshio Mimura, Kobe, Japan

## 3416

A cubic polynomial :

(X + 183^{2})(X + 288^{2})(X + 3416^{2}) = X^{3} + 3433^{2}X^{2} + 1166808^{2}X + 180036864^{2}.

The quadratic polynomial -3416X^{2} + 22624X - 10559 takes the values 93^{2}, 145^{2}, 163^{2}, 159^{2}, 131^{2}, 47^{2} at X = 1, 2,..., 6,

by Yoshio Mimura, Kobe, Japan

## 3417

3417 = (1^{2} + 2^{2} + 3^{2} + ... + 67^{2}) / (1^{2} + 3^{2} + 4^{2}).

by Yoshio Mimura, Kobe, Japan

## 3418

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

3418 - 1480 - 6596 - 13441 - ... - 4768 - 6833 - 5713 - 3418

(Note f(3418) = 34^{2} + 18^{2} = 1480, f(1480) = 14^{2} + 80^{2} = 6596, etc. See 1268)

by Yoshio Mimura, Kobe, Japan

## 3420

Komachi equations:

3420^{2} = 9^{2} * 8^{2} * 76^{2} * 5^{2} * 4^{2} / 32^{2} */ 1^{2} = 9^{2} / 8^{2} * 76^{2} * 5^{2} / 4^{2} * 32^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 3421

3421^{2} = 11703241, 1^{2} + 1^{2} + 7^{2} + 0^{2} + 3^{2} + 2^{2} + 4^{2} + 1^{2} = 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 3423

3423^{2} = 1302^{2} + 2142^{2} + 2331^{2} = 1332^{2} + 2412^{2} + 2031^{2}.

by Yoshio Mimura, Kobe, Japan

## 3424

3424^{2} = (1^{2} + 7)(10^{2} + 7)(117^{2} + 7) = (10^{2} + 7)(331^{2} + 7).

3424^{2} = 44^{3} + 116^{3} + 216^{3}.

(3424^{2} + 2) = (1^{2} + 2)(3^{2} + 2)(5^{2} + 2)(7^{2} + 2)(16^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 3425

3425^{2} = 10^{4} + 10^{4} + 40^{4} + 55^{4}.

Loop of length 5 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

3425 - 1781 - 6850 - 7124 - ... - 6850 - 7124 - 5617 - 3425

(Note f(3425)= 34^{2} + 25^{2} = 1781, f(1781) = 17^{2} + 81^{2} = 6850, etc. See 1781)

by Yoshio Mimura, Kobe, Japan

## 3426

The square root of 3426 is 58.532042...., and 58 = 5^{2} + 3^{2} + 2^{2} + 0^{2} + 4^{2} + 2^{2}.

by Yoshio Mimura, Kobe, Japan

## 3429

3429^{2} = 145^{3} + 156^{3} + 170^{3}.

1 / 3429 = 0.0002916...., and 2916 = 54^{2}.

by Yoshio Mimura, Kobe, Japan

## 3430

Komachi equations:

3430^{2} = 98^{2} * 7^{2} * 6^{2} * 5^{2} / 4^{2} / 3^{2} * 2^{2} */ 1^{2} = 98^{2} * 7^{2} / 6^{2} * 5^{2} * 4^{2} * 3^{2} / 2^{2} */ 1^{2}.

3430^{2} = (4 + 5 + 6 + 7 + 8 + 9 + 10)^{2} + (11 + 12 + 13 + 14 + 15 + 16 + 17)^{2} + (18 + 19 + 20 + 21 + 22 + 23 + 24)^{2} + ... + (165 + 166 + 167 + 168 + 169 + 170 + 171)^{2}.

938^{k} + 1274^{k} + 3430^{k} + 6902^{k} are squares for k = 1,2,3 (112^{2}, 7868^{2}, 609952^{2}).

by Yoshio Mimura, Kobe, Japan

## 3431

3431^{2} = 11771761, a square with just 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 3432

3432^{2}± 5 are primes.

3432^{2} = (12^{2} - 1)(287^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 3433

3433^{2} = 1^{4} + 28^{4} + 34^{4} + 56^{4} = 14^{4} + 14^{4} + 37^{4} + 56^{4}.

A cubic polynomial :

(X + 183^{2})(X + 288^{2})(X + 3416^{2}) = X^{3} + 3433^{2}X^{2} + 1166808^{2}X + 180036864^{2}.

by Yoshio Mimura, Kobe, Japan

## 3437

3437^{2} = S_{2}(182) + S_{2}(308), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + .... + n^{2}.

3437^{2} = 11812969, 118 * 1 * 29 + 6 + 9 = 3437.

294^{k} + 1288^{k} + 1617^{k} + 2730^{k} are squares for k = 1,2,3 (77^{2}, 3437^{2}, 163513^{2}).

by Yoshio Mimura, Kobe, Japan

## 3441

3441^{2} = 11840481, 1 - 1 + 840 * 4 + 81 = 1 * 1 * 840 * 4 + 81 = 3441.

3441^{k} + 6290^{k} + 10138^{k} + 14356^{k} are squares for k = 1,2,3 (185^{2}, 18981^{2}, 2071297^{2}).

by Yoshio Mimura, Kobe, Japan

## 3444

3444^{2}± 5 are primes.

3444^{2} = 14^{3} + 160^{3} + 198^{3}.

by Yoshio Mimura, Kobe, Japan

## 3445

3445 = (1^{2} + 2^{2} + 3^{2} + ... + 52^{2}) / (1^{2} + 2^{2} + 3^{2}).

3445^{2} = 11868025, 1 - 1 + 86 * 80 / 2 + 5 = 1 * 1 * 86 * 80 / 2 + 5 = 3445.

by Yoshio Mimura, Kobe, Japan

## 3446

3446^{2} = 11874916, 11 - 8 + 7 * 491 + 6 = 3446.

by Yoshio Mimura, Kobe, Japan

## 3447

3447^{2} = 49^{3} + 99^{3} + 221^{3}.

by Yoshio Mimura, Kobe, Japan

## 3449

1 / N = 0.000289...., and 289 = 17^{2} (N = 3449,..., 3460).

by Yoshio Mimura, Kobe, Japan

## 3450

The square root of 3450 is 58.73...., and 58 = 7^{2} + 3^{2}.

S_{2}(3450) = S_{2}(2669) + S_{2}(2804), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + .... + n^{2}.

3450^{2} = (83 + 84 + 85 + 86 + 87)^{2} + (88 + 89 + 90 + 91 + 92)^{2} + (93 + 94 + 95 + 96 + 97)^{2} + ... + (193 + 194 + 195 + 196 + 197)^{2}.

3450^{k} + 10833^{k} + 12972^{k} + 15594^{k} are squares for k = 1,2,3 (207^{2}, 23253^{2}, 2699487^{2}).

by Yoshio Mimura, Kobe, Japan

## 3451

1110^{2} + 3451^{2} = 13141501, a mosaic square.

by Yoshio Mimura, Kobe, Japan

## 3453

3453^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 3454

3454^{2} = 11930116, 119 * 30 - 116 = 1 - 1 * 9 + 577 * 6 + 4 = 1 * 1 - 9 + 577 * 6 + 4

= 11 + 9 * 64 * 6 - 8 * 1 = 3459.

by Yoshio Mimura, Kobe, Japan

## 3456

3456^{2} = 12^{6} + 12^{6} + 12^{6} + 12^{6}.

60^{2} + 3456 = 84^{2}, 60^{2} - 3456 = 12^{2}.

Komachi equations:

3456^{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}.

by Yoshio Mimura, Kobe, Japan

## 3458

3458^{2} = (2^{2} + 3)(1307^{2} + 3) = (4^{2} + 3)(7^{2} + 3)(110^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 3459

3459^{2} = 110^{3} + 172^{3} + 177^{3}.

by Yoshio Mimura, Kobe, Japan

## 3460

Loop of length 4 (See 41):

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

by Yoshio Mimura, Kobe, Japan

## 3462

3462^{2} = (1^{2} + 2)(2^{2} + 2)(816^{2} + 2) = (4^{2} + 2)(816^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 3463

3463^{2} = 24^{4} + 24^{4} + 41^{4} + 54^{4}.

by Yoshio Mimura, Kobe, Japan

## 3465

3465^{2} = (83 + 84 + 85)^{2} + (86 + 87 + 88)^{2} + (89 + 90 + 91)^{2} + ... + (230 + 231 + 232)^{2}.

3465^{2} = 91^{2} + 92^{2} + 93^{2} + 94^{2} + 95^{2} + 96^{2} + ... + 332^{2}.

3465^{2} = (2^{2} - 1)(6^{2} - 1)(10^{2} - 1)(34^{2} - 1).

18^{k} + 84^{k} + 129^{k} + 210^{k} are squares for k = 1,2,3 (21^{2}, 261^{2}, 3465^{2}).

by Yoshio Mimura, Kobe, Japan

## 3468

3468^{2} = 7^{3} + 86^{3} + 225^{3}.

by Yoshio Mimura, Kobe, Japan

## 3472

3472^{2} = 12054784 appears in the decimal expressions of π:

π = 3.14159•••12054784••• (from the 27922nd digit)

(12054784 is the third 8-digit square in the expression of π.)

by Yoshio Mimura, Kobe, Japan

## 3473

3473^{2} = 12061729, a zigzag square.

3473^{2} = 4^{4} + 19^{4} + 28^{4} + 58^{4}.

3473^{2} = 12061729, 1 * 206 * 17 - 29 = 3473.

3473^{4} = 145485306469441, and 1^{2} + 4^{2} + 5^{2} + 48^{2} + 5^{2} + 30^{2} + 6^{2} + 4^{2} + 6^{2} + 9^{2} + 4^{2} + 4^{2} + 1^{2} = 3473.

by Yoshio Mimura, Kobe, Japan

## 3474

3474^{5} = 505998314925198624 : 5^{2} + 0^{2} + 5^{2} + 9^{2} + 9^{2} + 8^{2} + 3^{2} + 1^{2} + 49^{2} + 2^{2} + 5^{2} + 1^{2} + 9^{2} + 8^{2} + 6^{2} + 24^{2} = 3474.

by Yoshio Mimura, Kobe, Japan

## 3475

3475^{2} = 12075625, a zigzag square.

3475^{2} = 12075625, 1^{2} + 2^{2} + 0^{2} + 7^{2} + 5^{2} + 6^{2} + 2^{2} + 5^{2} = 12^{2}.

by Yoshio Mimura, Kobe, Japan

## 3476

3476^{2} = (13^{2} + 7)(262^{2} + 7) = (2^{2} + 7)(3^{2} + 7)(262^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 3481

The square of 59.

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

## 3482

3482^{2} = 12124324, a square every digit of which is non-zero and smaller than 5.

by Yoshio Mimura, Kobe, Japan

## 3484

3484^{2} = 13^{3} + 76^{3} + 227^{3}.

3484^{2} = 12138256, 1^{2} + 2^{2} + 1^{2} + 3^{2} + 8^{2} + 2^{2} + 5^{2} + 6^{2} = 12^{2}.

3484^{2} = (7^{2} + 3)(8^{2} + 3)(59^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 3485

3485^{2} = (16 + 17 + 18 + ... + 56)^{2} + (57 + 58 + 59 + ... + 97)^{2} + (98 + 99 + 100 + ... + 138)^{2} + ... + (57 + 58 + 59 + ... + 97)^{2}.

3485^{2} = (2^{2} + 1)(4^{2} + 1)(378^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 3486

3486^{2} = (1^{2} + 5)(24^{2} + 5)(59^{2} + 5).

3486^{2} = 20^{3} + 91^{3} + 225^{3}.

by Yoshio Mimura, Kobe, Japan

## 3489

3489^{2} = 63^{3} + 149^{3} + 205^{3}.

by Yoshio Mimura, Kobe, Japan

## 3492

3492^{2} = 4^{3} + 30^{3} + 230^{3} = 24^{4} + 36^{4} + 36^{4} + 54^{4}.

3492^{5} = 519243627351647232 : 5^{2} + 19^{2} + 2^{2} + 43^{2} + 6^{2} + 2^{2} + 7^{2} + 3^{2} + 5^{2} + 1^{2} + 6^{2} + 4^{2} + 7^{2} + 2^{2} + 32^{2} = 3492.

by Yoshio Mimura, Kobe, Japan

## 3493

3493^{2} = 71^{3} + 91^{3} + 223^{3}.

by Yoshio Mimura, Kobe, Japan

## 3495

3495^{2} = 12215025, 1^{2} + 2^{2} + 2^{2} + 1^{2} + 5^{2} + 0^{2} + 2^{2} + 5^{2} = 8^{2}.

by Yoshio Mimura, Kobe, Japan

## 3497

2080^{k} + 3497^{k} + 4004^{k} + 4108^{k} are squares for k = 1,2,3 (117^{2}, 7033^{2}, 430443^{2}).

by Yoshio Mimura, Kobe, Japan

## 3498

3498^{2} = 9^{4} + 13^{4} + 17^{4} + 59^{4}.

by Yoshio Mimura, Kobe, Japan