## 41019

41019^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41038

41039^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41042

41042^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41056

41056^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41114

41114^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41125

41125 ^{2} = 1691265625, and 169 = 13^{2}, 1265625 = 1125^{2}.

by Yoshio Mimura, Kobe, Japan

## 41127

41127^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41139

41139^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41160

259^{2} + 41160 = 329^{2}, 259^{2} - 41160 = 161^{2},

203^{2} + 41160 = 287^{2}, 203^{2} - 41160 = 7^{2}.

by Yoshio Mimura, Kobe, Japan

## 41184

250^{2} + 41184 = 322^{2}, 250^{2} - 41184 = 146^{2}.

41184^{2} = (2^{2} - 1)(3^{2} - 1)(12^{2} - 1)(703^{2} - 1) = (2^{2} - 1)(7^{2} - 1)(12^{2} - 1)(287^{2} - 1)

= (3^{2} - 1)(5^{2} - 1)(10^{2} - 1)(12^{2} - 1)(25^{2} - 1) = (3^{2} - 1)(12^{2} - 1)(23^{2} - 1)(53^{2} - 1)

= (5^{2} - 1)(12^{2} - 1)(703^{2} - 1) = (12^{2} - 1)(53^{2} - 1)(65^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 41209

41209 = 203^{2}, a zigzag square with different digita.

by Yoshio Mimura, Kobe, Japan

## 41216

41216^{2} = (1^{2} + 7)(4^{2} + 7)(7^{2} + 7)(19^{2} + 7)(21^{2} + 7) = (3^{2} + 7)(7^{2} + 7)(21^{2} + 7)(65^{2} + 7)

= (4^{2} + 7)(5^{2} + 7)(7^{2} + 7)(203^{2} + 7) = (7^{2} + 7)(27^{2} + 7)(203^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 41225

41225 ^{2} = 1699500625, and 16 = 4^{2}, 99500625 = 9975^{2}.

by Yoshio Mimura, Kobe, Japan

## 41250

41250^{2} = (2^{2} + 6)(3^{2} + 6)(7^{2} + 6)(12^{2} + 6)(37^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 41258

41258^{2} = (2^{2} + 3)(41^{2} + 3)(380^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 41268

41268^{2} = (4^{2} + 3)(15^{2} + 3)(627^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 41262

41262^{2} = 3004 x 3005 + 3006 x 3007 + 3008 x 3009 + 3010 x 3011 + ... + 3340 x 3341.

by Yoshio Mimura, Kobe, Japan

## 41290

41290^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41323

(41323^{2} +1 ) = (1^{2} + 1)(2^{2} + 1)(4^{2} + 1)(6^{2} + 1)(20^{2} + 1)(26^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 41361

41361^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41363

41363^{2} = 1807^{2} + 1809^{2} + 1811^{2} + 1813^{2} + 1815^{2} + ... + 2527^{2}.

by Yoshio Mimura, Kobe, Japan

## 41376

41376^{2} = 1711973376, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 41382

41382^{2} = (1^{2} + 2)(2^{2} + 2)(3^{2} + 2)(6^{2} + 2)(19^{2} + 2)(25^{2} + 2)

= (1^{2} + 2)(2^{2} + 2)(8^{2} + 2)(19^{2} + 2)(63^{2} + 2) = (1^{2} + 2)(3^{2} + 2)(8^{2} + 2)(14^{2} + 2)(63^{2} + 2)

= (1^{2} + 2)(6^{2} + 2)(8^{2} + 2)(19^{2} + 2)(25^{2} + 2) = (2^{2} + 2)(3^{2} + 2)(4^{2} + 2)(19^{2} + 2)(63^{2} + 2)

= (2^{2} + 2)(14^{2} + 2)(19^{2} + 2)(63^{2} + 2) = (3^{2} + 2)(4^{2} + 2)(6^{2} + 2)(19^{2} + 2)(25^{2} + 2)

= (3^{2} + 2)(6^{2} + 2)(8^{2} + 2)(13^{2} + 2)(19^{2} + 2) = (4^{2} + 2)(8^{2} + 2)(19^{2} + 2)(63^{2} + 2)

= (6^{2} + 2)(14^{2} + 2)(19^{2} + 2)(25^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 41391

41391^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41400

41400^{2} = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(24^{2} - 1)(91^{2} - 1) = (2^{2} - 1)(11^{2} - 1)(24^{2} - 1)(91^{2} - 1)

= (4^{2} - 1)(5^{2} - 1)(24^{2} - 1)(91^{2} - 1) = (19^{2} - 1)(24^{2} - 1)(91^{2} - 1) = (1^{2} + 1)(2^{2} + 1)(10^{2} + 1)(1303^{2} + 1)

= (1^{2} + 1)(10^{2} + 1)(32^{2} + 1)(91^{2} + 1) = (3^{2} + 1)(10^{2} + 1)(1303^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 41404

41404^{2} = 2029^{2} + 2030^{2} + 2031^{2} + 2032^{2} + ... + 2380^{2}.

by Yoshio Mimura, Kobe, Japan

## 41410

41410^{2} = (1^{2} + 1)(10^{2} + 1)(32^{2} + 1)(91^{2} + 1)

= (1^{2} + 1)(2^{2} + 1)(10^{2} + 1)(1303^{2} + 1) = (3^{2} + 1)(10^{2} + 1)(1303^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 41414

41414^{2} = 1715119396, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 41442

41442^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 41444

41444^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41454

41454^{2} = (1^{2} + 5)(2^{2} + 5)(18^{2} + 5)(311^{2} + 5) = (1^{2} + 5)(18^{2} + 5)(29^{2} + 5)(32^{2} + 5)

= (4^{2} + 5)(17^{2} + 5)(18^{2} + 5)(29^{2} + 5) = (4^{2} + 5)(29^{2} + 5)(311^{2} + 5) = (7^{2} + 5)(18^{2} + 5)(311^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 41490

41490^{2} = (2^{2} + 5)(5^{2} + 5)(2525^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 41496

205^{2} + 41496 = 289^{2}, 205^{2} - 41496 = 23^{2}.

41496^{2} = (1^{2} + 3)(2^{2} + 3)(3^{2} + 3)(4^{2} + 3)(7^{2} + 3)(72^{2} + 3)

= (1^{2} + 3)(2^{2} + 3)(4^{2} + 3)(6^{2} + 3)(15^{2} + 3)(19^{2} + 3) = (1^{2} + 3)(2^{2} + 3)(6^{2} + 3)(9^{2} + 3)(137^{2} + 3)

= (1^{2} + 3)(2^{2} + 3)(7^{2} + 3)(15^{2} + 3)(72^{2} + 3) = (1^{2} + 3)(3^{2} + 3)(4^{2} + 3)(19^{2} + 3)(72^{2} + 3)

= (1^{2} + 3)(4^{2} + 3)(6^{2} + 3)(23^{2} + 3)(33^{2} + 3) = (1^{2} + 3)(4^{2} + 3)(7^{2} + 3)(9^{2} + 3)(72^{2} + 3)

= (1^{2} + 3)(15^{2} + 3)(19^{2} + 3)(72^{2} + 3) = (2^{2} + 3)(3^{2} + 3)(4^{2} + 3)(6^{2} + 3)(7^{2} + 3)(23^{2} + 3)

= (2^{2} + 3)(3^{2} + 3)(4^{2} + 3)(23^{2} + 3)(45^{2} + 3) = (2^{2} + 3)(3^{2} + 3)(5^{2} + 3)(6^{2} + 3)(137^{2} + 3)

= (2^{2} + 3)(3^{2} + 3)(33^{2} + 3)(137^{2} + 3) = (2^{2} + 3)(4^{2} + 3)(5^{2} + 3)(6^{2} + 3)(7^{2} + 3)(15^{2} + 3)

= (2^{2} + 3)(4^{2} + 3)(5^{2} + 3)(15^{2} + 3)(45^{2} + 3) = (2^{2} + 3)(4^{2} + 3)(7^{2} + 3)(15^{2} + 3)(33^{2} + 3)

= (2^{2} + 3)(6^{2} + 3)(7^{2} + 3)(15^{2} + 3)(23^{2} + 3) = (2^{2} + 3)(15^{2} + 3)(23^{2} + 3)(45^{2} + 3)

= (3^{2} + 3)(4^{2} + 3)(5^{2} + 3)(7^{2} + 3)(72^{2} + 3) = (3^{2} + 3)(4^{2} + 3)(6^{2} + 3)(7^{2} + 3)(61^{2} + 3)

= (3^{2} + 3)(4^{2} + 3)(6^{2} + 3)(19^{2} + 3)(23^{2} + 3) = (3^{2} + 3)(4^{2} + 3)(45^{2} + 3)(61^{2} + 3)

= (3^{2} + 3)(7^{2} + 3)(12^{2} + 3)(137^{2} + 3) = (3^{2} + 3)(7^{2} + 3)(15^{2} + 3)(110^{2} + 3)

= (3^{2} + 3)(7^{2} + 3)(23^{2} + 3)(72^{2} + 3) = (3^{2} + 3)(9^{2} + 3)(1307^{2} + 3)

= (4^{2} + 3)(5^{2} + 3)(6^{2} + 3)(15^{2} + 3)(19^{2} + 3) = (4^{2} + 3)(6^{2} + 3)(7^{2} + 3)(9^{2} + 3)(23^{2} + 3)

= (4^{2} + 3)(9^{2} + 3)(23^{2} + 3)(45^{2} + 3) = (4^{2} + 3)(15^{2} + 3)(19^{2} + 3)(33^{2} + 3)

= (5^{2} + 3)(6^{2} + 3)(9^{2} + 3)(137^{2} + 3) = (5^{2} + 3)(7^{2} + 3)(15^{2} + 3)(72^{2} + 3)

= (6^{2} + 3)(7^{2} + 3)(15^{2} + 3)(61^{2} + 3) = (6^{2} + 3)(15^{2} + 3)(19^{2} + 3)(23^{2} + 3)

= (9^{2} + 3)(33^{2} + 3)(137^{2} + 3) = (15^{2} + 3)(45^{2} + 3)(61^{2} + 3) = (20^{2} - 1)(27^{2} - 1)(77^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 41524

41524^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41544

41544^{2} = (1^{2} + 8)(8^{2} + 8)(1632^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 41580

41580^{2} = (2^{2} - 1)(4^{2} - 1)(8^{2} - 1)(23^{2} - 1)(34^{2} - 1) = (2^{2} - 1)(8^{2} - 1)(9^{2} - 1)(10^{2} - 1)(34^{2} - 1)

= (2^{2} - 1)(8^{2} - 1)(34^{2} - 1)(89^{2} - 1) = (2^{2} - 1)(10^{2} - 1)(34^{2} - 1)(71^{2} - 1) = (4^{2} - 1)(10^{2} - 1)(1079^{2} - 1)

= (4^{2} - 1)(6^{2} - 1)(8^{2} - 1)(10^{2} - 1)(23^{2} - 1) = (6^{2} - 1)(8^{2} - 1)(10^{2} - 1)(89^{2} - 1)

= (7^{2} - 1)(8^{2} - 1)(10^{2} - 1)(76^{2} - 1) = (10^{2} - 1)(55^{2} - 1)(76^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 41600

41600^{2} = (2^{2} + 4)(3^{2} + 4)(4^{2} + 4)(6^{2} + 4)(10^{2} + 4)(14^{2} + 4)

= (2^{2} + 4)(6^{2} + 4)(10^{2} + 4)(14^{2} + 4)(16^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 41607

41607^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41613

41613^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41616

41616^{2}± 5 are primes.

41616 = 204^{2}, a zigzag square pegged by 1.

41616^{2} = (1^{2} + 8)(2^{2} + 8)(3^{2} + 8)(20^{2} + 8)(48^{2} + 8) = (1^{2} + 8)(3^{2} + 8)(4^{2} + 8)(14^{2} + 8)(48^{2} + 8)

= (3^{2} + 8)(10^{2} + 8)(20^{2} + 8)(48^{2} + 8) = (1^{2} + 8)(14^{2} + 8)(20^{2} + 8)(48^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 41630

41630^{2} = (15^{2} + 5)(2745^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 41668

41668^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41688

41688^{2} = (4^{2} + 8)(68^{2} + 8)(125^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 41733

41733^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 41756

41756^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41760

41760^{2} = (3^{2} - 1)(9^{2} - 1)(28^{2} - 1)(59^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 41770

41770^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 41796

41796^{2} = (1^{2} + 8)(10^{2} + 8)(11^{2} + 8)(118^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 41810

41810^{2} = (4^{2} + 4)(9349^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 41832

41832^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 41847

41847^{2} = 1663^{2} + 1664^{2} + 1665^{2} + 1666^{2} + ... + 2143^{2}.

by Yoshio Mimura, Kobe, Japan

## 41854

41854^{2} = 1751757316, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 41873

41873^{2} = 2107^{2} + 2108^{2} + 2109^{2} + 2109^{2} + ... + 2444^{2}.

by Yoshio Mimura, Kobe, Japan

## 41876

41876^{2} = 1753599376, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 41930

41930^{2} = 3987 x 3988 + 3989 x 3990 + 3991 x 3992 + 3993 x 3994 + ... + 4195 x 4196.

by Yoshio Mimura, Kobe, Japan

## 41959

1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + 41959^{2} = 24624626864860, which consists of even digits.

by Yoshio Mimura, Kobe, Japan