## 8601

8601 = 10^{2} + 21^{2} + 32^{2} + ... + (11n + 10)^{2} + ... + 1341^{2}.

S_{2}(8601) = S_{2}(3484) + S_{2}(8406), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 8605

Loop of length 5 : 8605 --- 86^{2} + 05^{2} = 7421 --- 72^{2} + 41^{2} = 5917 --- 59^{2} + 17^{2} = 3770 --- 37^{2} + 70^{2} = 6269 --- 62^{2} + 69^{2} = 8605

by Yoshio Mimura, Kobe, Japan

## 8607

8607^{4} = 5487912924041601, and

5^{2} + 4^{2} + 87^{2} + 9^{2} + 1^{2} + 29^{2} + 2^{2} + 4^{2} + 0^{2} + 4^{2} + 1^{2} + 6^{2} + 0^{2} + 1^{2} = 8607.

by Yoshio Mimura, Kobe, Japan

## 8609

8609^{5} = 47289371176810582049, and

4^{2} + 7^{2} + 2^{2} + 8^{2} + 9^{2} + 3^{2} + 7^{2} + 1^{2} + 1^{2} + 76^{2} + 8^{2} + 1^{2} + 0^{2} + 5^{2} + 8^{2} + 2^{2} + 0^{2} + 49^{2} = 8609.

by Yoshio Mimura, Kobe, Japan

## 8610

8610^{2} = (3^{2} + 5)(6^{2} + 5)(10^{2} + 5)(35^{2} + 5).

8610^{2} = 409 x 410 + 411 x 412 + 413 x 414 + 415 x 416 + ... + 799 x 800.

by Yoshio Mimura, Kobe, Japan

## 8614

8614^{2} = 225^{3} + 315^{3} + 316^{3}.

8614^{5} = 47426856172874825824, and

4^{2} + 7^{2} + 4^{2} + 26^{2} + 8^{2} + 5^{2} + 6^{2} + 1^{2} + 7^{2} + 28^{2} + 7^{2} + 4^{2} + 8^{2} + 2^{2} + 5^{2} + 82^{2} + 4^{2}

= 4^{2} + 7^{2} + 4^{2} + 26^{2} + 8^{2} + 5^{2} + 6^{2} + 1^{2} + 7^{2} + 28^{2} + 7^{2} + 4^{2} + 82^{2} + 5^{2} + 8^{2} + 2^{2} + 4^{2} = 8614.

by Yoshio Mimura, Kobe, Japan

## 8616

8616^{2} = 74235456 appears in the decimal expressions of e:

e = 2.71828•••74235456••• (from the 21530th digit)

(74235456 is the fourth 8-digit square in the expression of e.)

by Yoshio Mimura, Kobe, Japan

## 8618

8618^{2} = S_{2}(462) + S_{2}(498), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 8621

8621^{5} = 47619872192762367101, and

47^{2} + 6^{2} + 1^{2} + 9^{2} + 8^{2} + 72^{2} + 1^{2} + 9^{2} + 27^{2} + 6^{2} + 2^{2} + 3^{2} + 6^{2} + 7^{2} + 10^{2} + 1^{2} = 8621.

by Yoshio Mimura, Kobe, Japan

## 8623

8623^{2} = 74356129, a square with different digits.

8623^{2} = 25^{3} + 267^{3} + 381^{3} = 26^{4} + 31^{4} + 34^{4} + 92^{4}.

by Yoshio Mimura, Kobe, Japan

## 8624

154^{k} + 8624^{k} + 17633^{k} + 26950^{k} are squares for k = 1,2,3 (231^{2}, 33341^{2}, 5069295^{2}).

by Yoshio Mimura, Kobe, Japan

## 8625

8625^{2} = 45^{3} + 295^{3} + 365^{3}.

8625^{2} = 74390625, a square with different digits.

8625^{5} = 47730448883056640625, and

4^{2} + 7^{2} + 73^{2} + 0^{2} + 4^{2} + 48^{2} + 8^{2} + 8^{2} + 3^{2} + 0^{2} + 5^{2} + 6^{2} + 6^{2} + 4^{2} + 0^{2} + 6^{2} + 25^{2} = 8625.

by Yoshio Mimura, Kobe, Japan

## 8626

1330^{k} + 8417^{k} + 8626^{k} + 10868^{k} are squares for k = 1,2,3 (171^{2}, 16283^{2}, 1588761^{2}).

by Yoshio Mimura, Kobe, Japan

## 8627

8627^{2} = 74425129, and 7^{2} + 4^{2} + 4^{2} + 2^{2} + 5^{2} + 1^{2} + 2^{2} + 9^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 8631

8631^{2} = 68^{3} + 238^{3} + 393^{3}.

by Yoshio Mimura, Kobe, Japan

## 8632

8632^{5} = 47924452309775122432, and

47^{2} + 9^{2} + 2^{2} + 44^{2} + 52^{2} + 30^{2} + 9^{2} + 7^{2} + 7^{2} + 5^{2} + 1^{2} + 2^{2} + 24^{2} + 3^{2} + 2^{2} = 8632.

by Yoshio Mimura, Kobe, Japan

## 8635

S_{2}(8635) = S_{2}(963) + S_{2}(8631), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 8637

8637^{2} = 233^{3} + 310^{3} + 318^{3}.

by Yoshio Mimura, Kobe, Japan

## 8640

8640^{2} = (2^{2} - 1)(3^{2} - 1)(11^{2} - 1)(161^{2} - 1) = (2^{2} - 1)(31^{2} - 1)(161^{2} - 1)

= (2^{2} - 1)(4^{2} - 1)(7^{2} - 1)(11^{2} - 1)(17^{2} - 1) = (3^{2} - 1)(19^{2} - 1)(161^{2} - 1)

= (3^{2} - 1)(4^{2} - 1)(5^{2} - 1)(161^{2} - 1) = (3^{2} - 1)(7^{2} - 1)(17^{2} - 1)(26^{2} - 1)

= (4^{2} - 1)(7^{2} - 1)(17^{2} - 1)(19^{2} - 1) = (5^{2} - 1)(11^{2} - 1)(161^{2} - 1).

102^{2} + 8640 = 138^{2}, 102^{2} - 8640 = 42^{2}.

8640^{2} = 164^{3} + 304^{3} + 348^{3}.

Komachi equations:

8640^{2} = 9^{2} * 8^{2} * 7^{2} * 6^{2} * 5^{2} * 4^{2} * 3^{2} / 21^{2} = 9^{2} * 8^{2} / 7^{2} * 6^{2} * 5^{2} * 4^{2} / 3^{2} * 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 8644

8644^{2} = 11^{3} + 308^{3} + 357^{3} = 32^{3} + 258^{3} + 386^{3} = 96^{3} + 170^{3} + 410^{3}.

1 / N = 0.0001156...., and 1156 = 34^{2}, for N = 8644, 8645,..., 8650.

by Yoshio Mimura, Kobe, Japan

## 8645

8645^{2} = 9^{3} + 140^{3} + 416^{3}.

8645^{5} = 48286417826561403125, and

48^{2} + 2^{2} + 8^{2} + 6^{2} + 4^{2} + 17^{2} + 8^{2} + 2^{2} + 65^{2} + 6^{2} + 1^{2} + 4^{2} + 0^{2} + 31^{2} + 25^{2}

= 48^{2} + 28^{2} + 64^{2} + 17^{2} + 8^{2} + 2^{2} + 6^{2} + 5^{2} + 6^{2} + 1^{2} + 4^{2} + 0^{2} + 31^{2} + 2^{2} + 5^{2} = 8645.

by Yoshio Mimura, Kobe, Japan

## 8646

8646^{2}± 5 are primes.

8646^{2} = 93^{3} + 242^{3} + 391^{3}.

by Yoshio Mimura, Kobe, Japan

## 8648

8648^{2} = 14^{4} + 62^{4} + 74^{4} + 74^{4}.

8648^{2} = 74787904, and 7^{2} + 4^{2} + 7^{2} + 8^{2} + 7^{2} + 9^{2} + 0^{2} + 4^{2} = 18^{2}.

by Yoshio Mimura, Kobe, Japan

## 8649

The square of 93.

403^{k} + 961^{k} + 5363^{k} + 8649^{k} are squares for k = 1,2,3 (124^{2}, 10230^{2}, 895652^{2}).

by Yoshio Mimura, Kobe, Japan

## 8652

8652^{2} = (2^{2} + 3)(3^{2} + 3)(10^{2} + 3)(93^{2} + 3) = (9^{2} + 3)(10^{2} + 3)(93^{2} + 3).

8652^{2} = 126^{3} + 262^{3} + 380^{3} = 202^{3} + 273^{3} + 359^{3} = 225^{3} + 311^{3} + 322^{3}.

8652^{5} = 48482226238696492032, and

4^{2} + 8^{2} + 4^{2} + 8^{2} + 2^{2} + 2^{2} + 26^{2} + 23^{2} + 8^{2} + 69^{2} + 6^{2} + 49^{2} + 2^{2} + 0^{2} + 3^{2} + 2^{2} = 8652.

by Yoshio Mimura, Kobe, Japan

## 8655

8655^{2} = 74909025, and 7^{2} + 4^{2} + 9^{2} + 0^{2} + 9^{2} + 0^{2} + 2^{2} + 5^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 8658

8658^{2} = 74960964, and 7 * 4 + 960 * 9 - 6 - 4 = 8658.

4836^{k} + 6825^{k} + 8658^{k} + 17706^{k} are squares for k = 1,2,3 (195^{2}, 21411^{2}, 2575053^{2}).

by Yoshio Mimura, Kobe, Japan

## 8659

8659^{2} = 74978281, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 8661

8661^{2} = 41^{3} + 250^{3} + 390^{3}.

8661^{5} = 48734912763123359301, and

4^{2} + 8^{2} + 7^{2} + 3^{2} + 4^{2} + 9^{2} + 12^{2} + 76^{2} + 31^{2} + 23^{2} + 3^{2} + 5^{2} + 9^{2} + 30^{2} + 1^{2} = 8661.

by Yoshio Mimura, Kobe, Japan

## 8662

8662^{5} = 48763053952327380832, and

4^{2} + 87^{2} + 6^{2} + 3^{2} + 0^{2} + 5^{2} + 3^{2} + 9^{2} + 5^{2} + 2^{2} + 3^{2} + 27^{2} + 3^{2} + 8^{2} + 0^{2} + 8^{2} + 3^{2} + 2^{2} = 8662.

by Yoshio Mimura, Kobe, Japan

## 8664

95^{2} + 8664 = 133^{2}, 95^{2} - 8664 = 19^{2}.

8664^{2} = 68^{3} + 304^{3} + 360^{3} = 152^{3} + 292^{3} + 360^{3}.

by Yoshio Mimura, Kobe, Japan

## 8667

8667^{2} = 88^{3} + 225^{3} + 398^{3}.

by Yoshio Mimura, Kobe, Japan

## 8668

8668^{4} = 5645151616082176, and

5^{2} + 6^{2} + 45^{2} + 15^{2} + 16^{2} + 16^{2} + 0^{2} + 8^{2} + 2^{2} + 1^{2} + 76^{2} = 8668.

by Yoshio Mimura, Kobe, Japan

## 8670

8670^{2} = 9^{3} + 322^{3} + 347^{3}.

by Yoshio Mimura, Kobe, Japan

## 8671

8671^{2} = 75186241, and 7^{2} + 5^{2} + 1^{2} + 8^{2} + 6^{2} + 2^{2} + 4^{2} + 1^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 8672

8672^{2} = 21^{3} + 35^{3} + 422^{3} = 120^{3} + 320^{3} + 344^{3}.

by Yoshio Mimura, Kobe, Japan

## 8673

8673^{2} = 49^{4} + 56^{4} + 56^{4} + 84^{4}.

8673^{2} = 1100^{2} + 1101^{2} + 1102^{2} + 1103^{2} + ... + 1158^{2}.

by Yoshio Mimura, Kobe, Japan

## 8676

8676^{2} = 75272976, and 75 * 2 + 7 * 29 * 7 * 6 = 8676.

by Yoshio Mimura, Kobe, Japan

## 8677

8677^{4} = 5668633640928241, and

5^{2} + 6^{2} + 6^{2} + 8^{2} + 6^{2} + 3^{2} + 3^{2} + 6^{2} + 40^{2} + 9^{2} + 2^{2} + 82^{2} + 4^{2} + 1^{2} = 8677.

8677^{5} = 49186734102334347157, and

49^{2} + 1^{2} + 8^{2} + 6^{2} + 7^{2} + 34^{2} + 10^{2} + 23^{2} + 3^{2} + 43^{2} + 47^{2} + 15^{2} + 7^{2}

= 49^{2} + 18^{2} + 6^{2} + 7^{2} + 3^{2} + 41^{2} + 0^{2} + 23^{2} + 34^{2} + 3^{2} + 47^{2} + 15^{2} + 7^{2} = 8677.

by Yoshio Mimura, Kobe, Japan

## 8679

8679^{2} = 1^{3} + 218^{3} + 402^{3} = 63^{3} + 77^{3} + 421^{3}.

by Yoshio Mimura, Kobe, Japan

## 8681

8681^{2} = 242^{3} + 272^{3} + 345^{3}.

by Yoshio Mimura, Kobe, Japan

## 8683

8683^{5} = 49357028460152043643, and

49^{2} + 3^{2} + 5^{2} + 70^{2} + 28^{2} + 4^{2} + 6^{2} + 0^{2} + 1^{2} + 5^{2} + 20^{2} + 4^{2} + 3^{2} + 6^{2} + 4^{2} + 3^{2} = 8683.

by Yoshio Mimura, Kobe, Japan

## 8685

8685^{2}± 2 are primes.

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

8685 - 14621 - 2558 - 3989 - ... - 9925 - 10426 - 693 - 8685

(Note f(8685) = 86^{2} + 85^{2} = 14621, f(14621) = 1^{2} + 46^{2} + 21^{2} = 2558, etc. See 37)

by Yoshio Mimura, Kobe, Japan

## 8686

8686^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 8687

8687^{2} = 32^{2} + 33^{2} + 34^{2} + 35^{2} + 36^{2} + ... + 609^{2}.

by Yoshio Mimura, Kobe, Japan

## 8688

8688^{2} = 75481344, and 7^{2} + 5^{2} + 4^{2} + 8^{2} + 1^{2} + 3^{2} + 4^{2} + 4^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 8689

8689^{2} = 75498721, and 7^{2} + 5^{2} + 4^{2} + 9^{2} + 8^{2} + 7^{2} + 2^{2} + 1^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 8694

8694^{2} = (1^{2} + 5)(2^{2} + 5)(3^{2} + 5)(8^{2} + 5)(38^{2} + 5) = (1^{2} + 5)(2^{2} + 5)(31^{2} + 5)(38^{2} + 5)

= (1^{2} + 5)(2^{2} + 5)(4^{2} + 5)(8^{2} + 5)(31^{2} + 5) = (1^{2} + 5)(8^{2} + 5)(11^{2} + 5)(38^{2} + 5)

= (2^{2} + 5)(17^{2} + 5)(169^{2} + 5) = (2^{2} + 5)(3^{2} + 5)(4^{2} + 5)(169^{2} + 5)

= (2^{2} + 5)(8^{2} + 5)(11^{2} + 5)(31^{2} + 5) = (3^{2} + 5)(38^{2} + 5)(61^{2} + 5)

= (3^{2} + 5)(4^{2} + 5)(8^{2} + 5)(61^{2} + 5) = (3^{2} + 5)(7^{2} + 5)(8^{2} + 5)(38^{2} + 5)

= (4^{2} + 5)(11^{2} + 5)(169^{2} + 5) = (4^{2} + 5)(31^{2} + 5)(61^{2} + 5) = (4^{2} + 5)(7^{2} + 5)(8^{2} + 5)(31^{2} + 5)

= (7^{2} + 5)(31^{2} + 5)(38^{2} + 5) = (8^{2} + 5)(17^{2} + 5)(61^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 8696

8696^{3} = 657595137536, and 6^{2} + 5^{2} + 75^{2} + 9^{2} + 5^{2} + 1^{2} + 3^{2} + 7^{2} + 53^{2} + 6^{2} = 8696.

by Yoshio Mimura, Kobe, Japan

## 8697

8697^{5} = 49756216673337223257, and

4^{2} + 9^{2} + 75^{2} + 6^{2} + 2^{2} + 16^{2} + 6^{2} + 7^{2} + 3^{2} + 3^{2} + 37^{2} + 2^{2} + 23^{2} + 25^{2} + 7^{2} = 8697.

by Yoshio Mimura, Kobe, Japan