## 40000

40000400001 = 200001^{2}.

by Yoshio Mimura, Kobe, Japan

## 40016

400016^{2} = (18^{2} + 4)(22^{2} + 4)(100^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 40032

40032^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 40040

40040^{2}± 3 are primes.

40040^{2} = (6^{2} - 1)(12^{2} - 1)(21^{2} - 1)(27^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 40054

40054^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40101

40101^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 40112

40112^{2} = (1^{2} + 7)(4^{2} + 7)(50^{2} + 7)(59^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 40150

40150^{2} = (2^{2} + 6)(7^{2} + 6)(1712^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 40166

40166^{2} = 3117 x 3118 + 3119 x 3120 + 3121 x 3122 + 3123 x 3124 + ... + 3417 x 3418.

by Yoshio Mimura, Kobe, Japan

## 40188

40188^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 40204

40204^{2} = 1616361616, a square with 3 kinds of digits 1,3,6.

40204^{2} = 1616361616, a square pegged by 6.

by Yoshio Mimura, Kobe, Japan

## 40282

40282^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40320

212^{2} + 40320 = 292^{2}, 212^{2} - 40320 = 68^{2}.

40320^{2} = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(5^{2} - 1)(15^{2} - 1)(29^{2} - 1)

= (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(11^{2} - 1)(13^{2} - 1)(15^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(5^{2} - 1)(6^{2} - 1)(7^{2} - 1)(41^{2} - 1)

= (2^{2} - 1)(3^{2} - 1)(5^{2} - 1)(6^{2} - 1)(15^{2} - 1)(19^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(5^{2} - 1)(17^{2} - 1)(99^{2} - 1)

= (2^{2} - 1)(3^{2} - 1)(6^{2} - 1)(11^{2} - 1)(127^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(29^{2} - 1)(41^{2} - 1)

= (2^{2} - 1)(3^{2} - 1)(9^{2} - 1)(13^{2} - 1)(71^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(15^{2} - 1)(19^{2} - 1)(29^{2} - 1)

= (2^{2} - 1)(4^{2} - 1)(7^{2} - 1)(9^{2} - 1)(97^{2} - 1) = (2^{2} - 1)(4^{2} - 1)(13^{2} - 1)(15^{2} - 1)(31^{2} - 1)

= (2^{2} - 1)(5^{2} - 1)(6^{2} - 1)(7^{2} - 1)(9^{2} - 1)(13^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(9^{2} - 1)(13^{2} - 1)(41^{2} - 1)

= (2^{2} - 1)(5^{2} - 1)(11^{2} - 1)(15^{2} - 1)(29^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(49^{2} - 1)(97^{2} - 1)

= (2^{2} - 1)(6^{2} - 1)(31^{2} - 1)(127^{2} - 1) = (2^{2} - 1)(7^{2} - 1)(9^{2} - 1)(13^{2} - 1)(29^{2} - 1)

= (3^{2} - 1)(4^{2} - 1)(5^{2} - 1)(6^{2} - 1)(127^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(7^{2} - 1)(13^{2} - 1)(41^{2} - 1)

= (3^{2} - 1)(4^{2} - 1)(8^{2} - 1)(15^{2} - 1)(31^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(13^{2} - 1)(15^{2} - 1)(19^{2} - 1)

= (3^{2} - 1)(4^{2} - 1)(29^{2} - 1)(127^{2} - 1) = (3^{2} - 1)(5^{2} - 1)(6^{2} - 1)(7^{2} - 1)(8^{2} - 1)(9^{2} - 1)

= (3^{2} - 1)(5^{2} - 1)(6^{2} - 1)(7^{2} - 1)(71^{2} - 1) = (3^{2} - 1)(5^{2} - 1)(6^{2} - 1)(9^{2} - 1)(55^{2} - 1)

= (3^{2} - 1)(5^{2} - 1)(6^{2} - 1)(17^{2} - 1)(29^{2} - 1) = (3^{2} - 1)(5^{2} - 1)(8^{2} - 1)(9^{2} - 1)(41^{2} - 1)

= (3^{2} - 1)(5^{2} - 1)(41^{2} - 1)(71^{2} - 1) = (3^{2} - 1)(6^{2} - 1)(11^{2} - 1)(13^{2} - 1)(17^{2} - 1)

= (3^{2} - 1)(6^{2} - 1)(15^{2} - 1)(161^{2} - 1) = (3^{2} - 1)(6^{2} - 1)(19^{2} - 1)(127^{2} - 1)

= (3^{2} - 1)(7^{2} - 1)(8^{2} - 1)(9^{2} - 1)(29^{2} - 1) = (3^{2} - 1)(7^{2} - 1)(29^{2} - 1)(71^{2} - 1)

= (3^{2} - 1)(9^{2} - 1)(29^{2} - 1)(55^{2} - 1) = (4^{2} - 1)(5^{2} - 1)(11^{2} - 1)(13^{2} - 1)(15^{2} - 1)

= (4^{2} - 1)(6^{2} - 1)(7^{2} - 1)(15^{2} - 1)(17^{2} - 1) = (4^{2} - 1)(15^{2} - 1)(17^{2} - 1)(41^{2} - 1)

= (5^{2} - 1)(6^{2} - 1)(11^{2} - 1)(127^{2} - 1) = (5^{2} - 1)(7^{2} - 1)(29^{2} - 1)(41^{2} - 1)

= (5^{2} - 1)(9^{2} - 1)(13^{2} - 1)(71^{2} - 1) = (5^{2} - 1)(15^{2} - 1)(19^{2} - 1)(29^{2} - 1)

= (6^{2} - 1)(13^{2} - 1)(17^{2} - 1)(31^{2} - 1) = (7^{2} - 1)(8^{2} - 1)(15^{2} - 1)(49^{2} - 1)

= (7^{2} - 1)(11^{2} - 1)(13^{2} - 1)(41^{2} - 1) = (7^{2} - 1)(13^{2} - 1)(449^{2} - 1) = (8^{2} - 1)(11^{2} - 1)(15^{2} - 1)(31^{2} - 1)

= (11^{2} - 1)(13^{2} - 1)(15^{2} - 1)(19^{2} - 1) = (11^{2} - 1)(29^{2} - 1)(127^{2} - 1) = (15^{2} - 1)(49^{2} - 1)(55^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 40330

40330^{2} = (1^{2} + 1)(6^{2} + 1)(33^{2} + 1)(142^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 40334

40334^{2} = (2^{2} + 3)(8^{2} + 3)(13^{2} + 3)(142^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 40341

40341^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 40344

205^{2} + 40344 = 287^{2}, 205^{2} - 40344 = 41^{2}.

by Yoshio Mimura, Kobe, Japan

## 40391

40391^{2} = 1 + 2 + 3 + 4 + ... + 57121 = 1^{2} - 2^{2} + 3^{2} - 4^{2} + ... + 57121^{2}.

40391^{2} is the sum of (2x + 1)^{3}, where 0 < 2x + 1 <= 337.

by Yoshio Mimura, Kobe, Japan

## 40392

40392^{2}± 5 are primes.

40392^{2} = (1^{2} + 8)(3^{2} + 8)(4^{2} + 8)(5^{2} + 8)(116^{2} + 8) = (1^{2} + 8)(3^{2} + 8)(5^{2} + 8)(20^{2} + 8)(28^{2} + 8)

= (1^{2} + 8)(3^{2} + 8)(28^{2} + 8)(116^{2} + 8) = (1^{2} + 8)(5^{2} + 8)(20^{2} + 8)(116^{2} + 8)

= (1^{2} + 8)(16^{2} + 8)(17^{2} + 8)(48^{2} + 8) = (1^{2} + 8)(48^{2} + 8)(280^{2} + 8)

= (2^{2} - 1)(10^{2} - 1)(35^{2} - 1)(67^{2} - 1) = (3^{2} + 8)(4^{2} + 8)(5^{2} + 8)(17^{2} + 8)(20^{2} + 8)

= (3^{2} + 8)(4^{2} + 8)(17^{2} + 8)(116^{2} + 8) = (3^{2} + 8)(4^{2} + 8)(28^{2} + 8)(71^{2} + 8)

= (3^{2} + 8)(8^{2} + 8)(16^{2} + 8)(71^{2} + 8) = (3^{2} + 8)(17^{2} + 8)(20^{2} + 8)(28^{2} + 8)

= (4^{2} + 8)(5^{2} + 8)(20^{2} + 8)(71^{2} + 8) = (4^{2} + 8)(71^{2} + 8)(116^{2} + 8)

= (5^{2} + 8)(8^{2} + 8)(17^{2} + 8)(48^{2} + 8) = (17^{2} + 8)(20^{2} + 8)(116^{2} + 8)

= (20^{2} + 8)(28^{2} + 8)(71^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 40401

40401 = 201^{2}, a zigzag square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 40404

40404^{2} = (1^{2} + 3)(2^{2} + 3)(6^{2} + 3)(21^{2} + 3)(58^{2} + 3) = (1^{2} + 3)(3^{2} + 3)(6^{2} + 3)(16^{2} + 3)(58^{2} + 3)

= (1^{2} + 3)(6^{2} + 3)(16^{2} + 3)(201^{2} + 3) = (2^{2} + 3)(6^{2} + 3)(7^{2} + 3)(16^{2} + 3)(21^{2} + 3)

= (2^{2} + 3)(16^{2} + 3)(21^{2} + 3)(45^{2} + 3) = (3^{2} + 3)(58^{2} + 3)(201^{2} + 3)

= (5^{2} + 3)(6^{2} + 3)(21^{2} + 3)(58^{2} + 3) = (6^{2} + 3)(16^{2} + 3)(19^{2} + 3)(21^{2} + 3)

= (21^{2} + 3)(33^{2} + 3)(58^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 40412

40412^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40414

40414^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40425

40425^{2} = (1^{2} + 6)(15^{2} + 6)(27^{2} + 6)(37^{2} + 6) = (3^{2} + 6)(13^{2} + 6)(27^{2} + 6)(29^{2} + 6)

= (7^{2} + 6)(13^{2} + 6)(15^{2} + 6)(27^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 40440

40440^{2}= 921 x 922 + 922 x 923 + 923 x 924 + 925 x 926 + ... + 1784 x 1785.

by Yoshio Mimura, Kobe, Japan

## 40467

40467^{2} = (4^{2} + 5)(6^{2} + 5)(18^{2} + 5)(76^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 40492

40492^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40495

4049540496 = 63636^{2}.

by Yoshio Mimura, Kobe, Japan

## 40501

40501^{2} = (10^{2} + 1)(4030^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 40545

40545^{2} = 1643897025, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 40557

40557^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 40560

221^{2} + 40560 = 299^{2}, 221^{2} - 40560 = 91^{2}.

by Yoshio Mimura, Kobe, Japan

## 40566

40566^{2} = (1^{2} + 5)(16561^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 40572

40572^{2} = (1^{2} + 5)(3^{2} + 5)(8^{2} + 5)(17^{2} + 5)(31^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 40584

40584^{2} = (7^{2} + 8)(12^{2} + 8)(436^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 40624

40624^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40636

40636^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40646

40646^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40656

275^{2} + 40656 = 341^{2}, 275^{2} - 40656 = 187^{2}.

by Yoshio Mimura, Kobe, Japan

## 40679

40679^{2} = 1139^{2} + 1140^{2} + 1141^{2} + ... + 1860^{2}.

by Yoshio Mimura, Kobe, Japan

## 40689

40689^{2} = (1^{2} + 8)(17^{2} + 8)(787^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 40700

40700^{2} = 1811^{2} + 1812^{2} + 1813^{2} + ... + 2217^{2}.

by Yoshio Mimura, Kobe, Japan

## 40718

40718^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40754

40754^{2}= 27 x 28 + 28 x 29 + 29 x 30 + 31 x 32 + ... + 1707 x 1708.

by Yoshio Mimura, Kobe, Japan

## 40776

40776^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 40800

40800^{2} = (3^{2} - 1)(9^{2} - 1)(16^{2} - 1)(101^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 40804

40804 = 202^{2}, a palindromic and zigzag square with even digits.

by Yoshio Mimura, Kobe, Japan

## 40810

40810^{2} = (1^{2} + 6)(4^{2} + 6)(7^{2} + 6)(10^{2} + 6)(43^{2} + 6) = (1^{2} + 6)(7^{2} + 6)(10^{2} + 6)(202^{2} + 6)

= (2^{2} + 6)(7^{2} + 6)(10^{2} + 6)(169^{2} + 6) = (2^{2} + 6)(10^{2} + 6)(29^{2} + 6)(43^{2} + 6)

= (4^{2} + 6)(43^{2} + 6)(202^{2} + 6) = (10^{2} + 6)(43^{2} + 6)(92^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 40824

40824^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 40832

40832^{2} = (1^{2} + 7)(2^{2} + 7)(43^{2} + 7)(101^{2} + 7) = (1^{2} + 7)(9^{2} + 7)(15^{2} + 7)(101^{2} + 7)

= (9^{2} + 7)(43^{2} + 7)(101^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 40836

40836^{2} = (18^{2} + 8)(22^{2} + 8)(101^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 40885

40885^{2} = (3^{2} + 4)(9^{2} + 4)(25^{2} + 4)(49^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 40920

40920^{2} = (21^{2} - 1)(32^{2} - 1)(61^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 40934

40934^{2} = 1679^{2} + 1681^{2} + 1683^{2} + 1685^{2} + 1687^{2} + ... + 2453^{2}.

by Yoshio Mimura, Kobe, Japan

## 40942

40942^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 40956

40956^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 40962

40962^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan