## 37000

37000^{2} = (1^{2} + 4)(6^{2} + 4)(11^{2} + 4)(234^{2} + 4) = (11^{2} + 4)(14^{2} + 4)(234^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 37026

37026^{2} = (1^{2} + 2)(3^{2} + 2)(14^{2} + 2)(19^{2} + 2)(24^{2} + 2) = (3^{2} + 2)(7^{2} + 2)(8^{2} + 2)(10^{2} + 2)(19^{2} + 2)

= (3^{2} + 2)(10^{2} + 2)(19^{2} + 2)(58^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 37034

37034^{2} = 1371517156, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 37044

37044^{2} = (1^{2} + 5)(3^{2} + 5)(7^{2} + 5)(17^{2} + 5)(32^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 37059

37059^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37060

37060^{2} = (1^{2} + 1)(3^{2} + 1)(33^{2} + 1)(251^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 37086

37086^{2} = 1375371396, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 37094

37094^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37134

37134^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 37149

37149^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37156

37156^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37158

37158^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 37164

37164^{2} = 808^{2} + 809^{2} + 810^{2} + ... + 1671^{2}.

by Yoshio Mimura, Kobe, Japan

## 37176

37176^{2} = 1382054976, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 37204

37204^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37246

37246^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37249

37249 = 193^{2}, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 37260

37260^{2} = (1^{2} + 5)(2^{2} + 5)(5^{2} + 5)(7^{2} + 5)(8^{2} + 5)(15^{2} + 5)

= (1^{2} + 5)(2^{2} + 5)(5^{2} + 5)(15^{2} + 5)(61^{2} + 5) = (5^{2} + 5)(7^{2} + 5)(15^{2} + 5)(61^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 37281

37281^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37292

37292^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37296

37296^{2} = (2^{2} - 1)(73^{2} - 1)(295^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 37312

37312^{2} = (1^{2} + 7)(2^{2} + 7)(5^{2} + 7)(24^{2} + 7)(29^{2} + 7) = (1^{2} + 7)(29^{2} + 7)(453^{2} + 7)

= (3^{2} + 7)(13^{2} + 7)(24^{2} + 7)(29^{2} + 7) = (5^{2} + 7)(9^{2} + 7)(24^{2} + 7)(29^{2} + 7)

= (24^{2} + 7)(29^{2} + 7)(53^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 37338

37338^{2} = (3^{2} + 5)(9979^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 37345

37345^{2} = 568^{2} + 569^{2} + 570^{2} + ... + 1634^{2}.

by Yoshio Mimura, Kobe, Japan

## 37352

37352^{2}± 3 are primes.

37352^{2} = (4^{2} + 7)(7^{2} + 7)(14^{2} + 7)(73^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 37431

37431^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37440

194^{2} + 37440 = 274^{2}, 194^{2} - 37440 = 14^{2}.

37440^{2} = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(25^{2} - 1)(79^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(14^{2} - 1)(79^{2} - 1)

= (2^{2} - 1)(7^{2} - 1)(9^{2} - 1)(14^{2} - 1)(25^{2} - 1) = (2^{2} - 1)(11^{2} - 1)(25^{2} - 1)(79^{2} - 1)

= (2^{2} - 1)(17^{2} - 1)(25^{2} - 1)(51^{2} - 1) = (3^{2} - 1)(5^{2} - 1)(51^{2} - 1)(53^{2} - 1)

= (4^{2} - 1)(5^{2} - 1)(25^{2} - 1)(79^{2} - 1) = (5^{2} - 1)(7^{2} - 1)(14^{2} - 1)(79^{2} - 1)

= (9^{2} - 1)(53^{2} - 1)(79^{2} - 1) = (19^{2} - 1)(25^{2} - 1)(79^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 37444

37444^{2} = (3^{2} + 7)(20^{2} + 7)(464^{2} + 7) = (4^{2} + 7)(20^{2} + 7)(387^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 37450

37450^{2} = (2^{2} + 6)(23^{2} + 6)(512^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 37464

37464^{2} = (1^{2} + 3)(3^{2} + 3)(9^{2} + 3)(590^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 37473

37473^{2} = 1404225729, with 1404225 = 1185^{2} and 729 = 27^{2}.

by Yoshio Mimura, Kobe, Japan

## 37485

37485^{2} = (6^{2} - 1)(8^{2} - 1)(16^{2} - 1)(50^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 37500

37500^{2} = (1^{3} + 2^{3} + 3^{3} + ... + 24^{3})(25^{3}).

by Yoshio Mimura, Kobe, Japan

## 37515

37515^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37521

37521^{2} = (1^{2} + 2)(3^{2} + 2)(5^{2} + 2)(1257^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 37536

37536737536 = 193744^{2}.

by Yoshio Mimura, Kobe, Japan

## 37544

37544^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37557

37557^{2} = 1872^{2} + 1873^{2} + 1874^{2} + ... + 2209^{2}.

by Yoshio Mimura, Kobe, Japan

## 37569

The quadratic polynomial 37569X^{2} - 275814X + 514921 takes the values 526^{2}, 337^{2}, 160^{2}, 113^{2}, 274^{2}, 461^{2}, 652^{2} at X = 1, 2,..., 7,

by Yoshio Mimura, Kobe, Japan

## 37570

37570^{2}= 5184 x 5185 + 5185 x 5186 + 5186 x 5187 +...+ 5235 x 5236.

37570^{2} = (1^{2} + 1)(4^{2} + 1)(8^{2} + 1)(21^{2} + 1)(38^{2} + 1) = (2^{2} + 1)(4^{2} + 1)(5^{2} + 1)(21^{2} + 1)(38^{2} + 1)

= (21^{2} + 1)(38^{2} + 1)(47^{2} + 1) = (4^{2} + 1)(5^{2} + 1)(38^{2} + 1)(47^{2} + 1) = (1^{2} + 9)(11^{2} + 9)(1042^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 37587

37587^{2} = (5^{2} + 8)(27^{2} + 8)(241^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 37598

37598^{2} = 1203^{2} + 1205^{2} + 1207^{2} + 1209^{2} + ... + 2169^{2}.

by Yoshio Mimura, Kobe, Japan

## 37599

37599^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37620

37620^{2} = (2^{2} - 1)(10^{2} - 1)(39^{2} - 1)(56^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 37636

37636 = 194^{2} is a square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 37638

37638^{2} = (1^{2} + 2)(2^{2} + 2)(5^{2} + 2)(24^{2} + 2)(71^{2} + 2) - (1^{2} + 2)(2^{2} + 2)(7^{2} + 2)(11^{2} + 2)(112^{2} + 2)

= (1^{2} + 2)(4^{2} + 2)(7^{2} + 2)(10^{2} + 2)(71^{2} + 2) = (1^{2} + 2)(10^{2} + 2)(11^{2} + 2)(194^{2} + 2)

= (1^{2} + 2)(112^{2} + 2)(194^{2} + 2) = (2^{2} + 2)(7^{2} + 2)(11^{2} + 2)(194^{2} + 2)

= (4^{2} + 2)(5^{2} + 2)(24^{2} + 2)(71^{2} + 2) = (4^{2} + 2)(7^{2} + 2)(11^{2} + 2)(112^{2} + 2)

= (22^{2} + 2)(24^{2} + 2)(71^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 37642

37642^{2} = 2651 x 2652 + 2653 x 2654 + 2655 x 2656 + 2657 x 2658 + ... + 3003 x 3004.

by Yoshio Mimura, Kobe, Japan

## 37664

37664^{2} = (1^{2} + 7)(2^{2} + 7)(3^{2} + 7)(10^{2} + 7)(97^{2} + 7) = (1^{2} + 7)(10^{2} + 7)(13^{2} + 7)(97^{2} + 7)

= (2^{2} + 7)(9^{2} + 7)(10^{2} + 7)(117^{2} + 7) = (2^{2} + 7)(10^{2} + 7)(11^{2} + 7)(97^{2} + 7)

= (2^{2} + 7)(97^{2} + 7)(117^{2} + 7) = (3^{2} + 7)(9^{2} + 7)(10^{2} + 7)(97^{2} + 7) = (10^{2} + 7)(31^{2} + 7)(117^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 37700

37700^{2} = (1^{2} + 1)(2^{2} + 1)(3^{2} + 1)(5^{2} + 1)(18^{2} + 1)(41^{2} + 1) = (1^{2} + 1)(2^{2} + 1)(5^{2} + 1)(41^{2} + 1)(57^{2} + 1)

= (1^{2} + 1)(2^{2} + 1)(5^{2} + 1)(7^{2} + 1)(8^{2} + 1)(41^{2} + 1) = (1^{2} + 1)(2^{2} + 1)(7^{2} + 1)(17^{2} + 1)(99^{2} + 1)

= (1^{2} + 1)(3^{2} + 1)(5^{2} + 1)(8^{2} + 1)(12^{2} + 1)(17^{2} + 1) = (1^{2} + 1)(3^{2} + 1)(7^{2} + 1)(12^{2} + 1)(99^{2} + 1)

= (1^{2} + 1)(3^{2} + 1)(7^{2} + 1)(17^{2} + 1)(70^{2} + 1) = (1^{2} + 1)(5^{2} + 1)(17^{2} + 1)(307^{2} + 1)

= (1^{2} + 1)(5^{2} + 1)(7^{2} + 1)(18^{2} + 1)(41^{2} + 1) = (3^{2} + 1)(5^{2} + 1)(41^{2} + 1)(57^{2} + 1)

= (3^{2} + 1)(5^{2} + 1)(7^{2} + 1)(8^{2} + 1)(41^{2} + 1) = (3^{2} + 1)(7^{2} + 1)(17^{2} + 1)(99^{2} + 1)

= (1^{2} + 4)(3^{2} + 4)(5^{2} + 4)(24^{2} + 4)(36^{2} + 4) = (1^{2} + 4)(4^{2} + 4)(5^{2} + 4)(24^{2} + 4)(29^{2} + 4)

= (4^{2} + 4)(5^{2} + 4)(11^{2} + 4)(140^{2} + 4) = (11^{2} + 4)(24^{2} + 4)(140^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 37739

37739^{2} = 1424232121, a square every digit of which is non-zero and smaller than 5.

by Yoshio Mimura, Kobe, Japan

## 37758

37758^{2} = (5^{2} + 6)(9^{2} + 6)(20^{2} + 6)(36^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 37779

37779^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37800

37800^{2} = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(6^{2} - 1)(13^{2} - 1)(26^{2} - 1)

= (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(8^{2} - 1)(251^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(6^{2} - 1)(26^{2} - 1)(29^{2} - 1)

= (2^{2} - 1)(6^{2} - 1)(11^{2} - 1)(13^{2} - 1)(26^{2} - 1) = (2^{2} - 1)(8^{2} - 1)(11^{2} - 1)(251^{2} - 1)

= (3^{2} - 1)(6^{2} - 1)(8^{2} - 1)(11^{2} - 1)(26^{2} - 1) = (4^{2} - 1)(5^{2} - 1)(6^{2} - 1)(13^{2} - 1)(26^{2} - 1)

= (4^{2} - 1)(5^{2} - 1)(8^{2} - 1)(251^{2} - 1) = (4^{2} - 1)(6^{2} - 1)(8^{2} - 1)(11^{2} - 1)(19^{2} - 1)

= (4^{2} - 1)(13^{2} - 1)(26^{2} - 1)(29^{2} - 1) = (6^{2} - 1)(8^{2} - 1)(26^{2} - 1)(31^{2} - 1)

= (6^{2} - 1)(13^{2} - 1)(19^{2} - 1)(26^{2} - 1) = (8^{2} - 1)(19^{2} - 1)(251^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 37829

37829^{2} = 1805^{2} + 1806^{2} + 1807^{2} + ... + 2166^{2}.

by Yoshio Mimura, Kobe, Japan

## 37830

37830^{2} = (9^{2} + 9)(31^{2} + 9)(128^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 37832

37832^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37842

37842^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 37880

37880^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37893

37893^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 37898

37898^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37905

37905^{2} = 1436789025, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 37906

37906^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37926

37926^{2} = (1^{2} + 5)(9^{2} + 5)(32^{2} + 5)(52^{2} + 5) = (3^{2} + 5)(9^{2} + 5)(32^{2} + 5)(34^{2} + 5)

= (4^{2} + 5)(9^{2} + 5)(17^{2} + 5)(52^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 37960

37960^{2} = (3^{2} + 4)(10^{2} + 4)(19^{2} + 4)(54^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 37961

37961^{2} = 1! + 6! + 8! + 9! + 10! + 12! + 12! + 12!.

by Yoshio Mimura, Kobe, Japan

## 37962

37962^{2} = 1441113444, a square with 3 kinds of digits 1,3,4.

by Yoshio Mimura, Kobe, Japan

## 37972

37972^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 37999

S2(37999) = S2(55) * S2(987), where S2(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan