## 39026

39026^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39102

39102^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 39116

39116^{2} = (2^{2} + 7)(35^{2} + 7)(336^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 39130

39130^{2} = 379 x 380 + 381 x 382 + 383 x 384 + 385 x 386 + ... + 2097 x 2098.

by Yoshio Mimura, Kobe, Japan

## 39147

39147^{2} = 1532487609, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 39150

39150^{2} = (1^{2} + 9)(6^{2} + 9)(36^{2} + 9)(51^{2} + 9) = (3^{2} + 9)(4^{2} + 9)(6^{2} + 9)(7^{2} + 9)(36^{2} + 9)

= (3^{2} + 9)(4^{2} + 9)(36^{2} + 9)(51^{2} + 9) = (4^{2} + 9)(6^{2} + 9)(9^{2} + 9)(123^{2} + 9)

= (6^{2} + 9)(7^{2} + 9)(21^{2} + 9)(36^{2} + 9) = (21^{2} + 9)(36^{2} + 9)(51^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 39166

39166^{2} = 1533975556, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 39168

39168^{2} = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(33^{2} - 1)(35^{2} - 1) = (5^{2} - 1)(7^{2} - 1)(33^{2} - 1)(35^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 39174

39174^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 39177

39177^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 39190

39190^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39200

39200^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39202

(39202^{2} - 4) = (5^{2} - 4)(8^{2} - 4)(9^{2} - 4)(10^{2} - 4)(13^{2} - 4)

= (3^{2} - 4)(4^{2} - 4)(5^{2} - 4)(9^{2} - 4)(10^{2} - 4)(13^{2} - 4).

by Yoshio Mimura, Kobe, Japan

## 39204

39204 = 198^{2}, a square with different digits.

39204^{2} = (1^{2} + 2)(2^{2} + 2)(4^{2} + 2)(8^{2} + 2)(14^{2} + 2)(19^{2} + 2)

= (1^{2} + 8)(2^{2} + 8)(5^{2} + 8)(17^{2} + 8)(38^{2} + 8) = (2^{2} + 2)(3^{2} + 2)(4^{2} + 2)(8^{2} + 2)(140^{2} + 2)

= (2^{2} + 2)(8^{2} + 2)(14^{2} + 2)(140^{2} + 2) = (5^{2} + 8)(10^{2} + 8)(17^{2} + 8)(38^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 39208

39208^{2} = (3^{2} + 4)(5^{2} + 4)(10^{2} + 4)(198^{2} + 4) = (82^{2} + 4)(478^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 39214

39214^{2} = 1537737796, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 39216

39216^{2} = (1^{2} + 3)(3^{2} + 3)(4^{2} + 3)(13^{2} + 3)(99^{2} + 3) = (1^{2} + 3)(13^{2} + 3)(15^{2} + 3)(99^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 39220

39220^{2} = (4^{2} + 4)(7^{2} + 4)(12^{2} + 4)(99^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 39250

39250^{2} = 1540562500, with 1 = 1^{2} and 540562500 = 23250^{2}.

39250^{2} = (1^{2} + 1)(2^{2} + 1)(28^{2} + 1)(443^{2} + 1) = (3^{2} + 1)(28^{2} + 1)(443^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 39260

39260^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39274

39274^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39312

39312^{2}± 5 are primes.

39312^{2} = (27^{2} - 1)(1457^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 39317

the sum of (13x + 5)^{2} = 1248665^{2}, where 13x + 5 <= 39317.

by Yoshio Mimura, Kobe, Japan

## 39336

39336^{2} = 1547320896, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 39338

39338^{2} = (21^{2} + 1)(34^{2} + 1)(55^{2} + 1) = (4^{2} + 1)(5^{2} + 1)(34^{2} + 1)(55^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 39358

39358^{2} = 4152^{2} + 4153^{2} + 4154^{2} + ... + 4239^{2}.

by Yoshio Mimura, Kobe, Japan

## 39368

39368^{2} = (1^{2} + 3)(4^{2} + 3)(5^{2} + 3)(16^{2} + 3)(53^{2} + 3) = (1^{2} + 3)(16^{2} + 3)(23^{2} + 3)(53^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 39376

39376^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39380

39380^{2} = 1921^{2} + 1922^{2} + 1923^{2} + ... + 2272^{2}.

by Yoshio Mimura, Kobe, Japan

## 39388

39388^{2} = 1551414544, a square with 3 kinds of digits 1,4,5.

by Yoshio Mimura, Kobe, Japan

## 39396

39396^{2} = (1^{2} + 3)(2^{2} + 3)(8^{2} + 3)(12^{2} + 3)(75^{2} + 3) = (1^{2} + 5)(3^{2} + 5)(53^{2} + 5)(81^{2} + 5)

= (5^{2} + 3)(8^{2} + 3)(12^{2} + 3)(75^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 39400

39400^{2} = (1^{2} + 4)(6^{2} + 4)(2786^{2} + 4) = (14^{2} + 4)(2786^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 39423

39423^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 39424

39424^{2} = (1^{2} + 7)(2^{2} + 7)(5^{2} + 7)(21^{2} + 7)(35^{2} + 7) = (1^{2} + 7)(3^{2} + 7)(7^{2} + 7)(13^{2} + 7)(35^{2} + 7)

= (1^{2} + 7)(5^{2} + 7)(7^{2} + 7)(9^{2} + 7)(35^{2} + 7) = (1^{2} + 7)(7^{2} + 7)(35^{2} + 7)(53^{2} + 7)

= (2^{2} + 7)(3^{2} + 7)(7^{2} + 7)(11^{2} + 7)(35^{2} + 7) = (2^{2} + 7)(5^{2} + 7)(7^{2} + 7)(13^{2} + 7)(21^{2} + 7)

= (2^{2} + 7)(7^{2} + 7)(21^{2} + 7)(75^{2} + 7) = (3^{2} + 7)(13^{2} + 7)(21^{2} + 7)(35^{2} + 7)

= (5^{2} + 7)(9^{2} + 7)(21^{2} + 7)(35^{2} + 7) = (7^{2} + 7)(11^{2} + 7)(13^{2} + 7)(35^{2} + 7)

= (21^{2} + 7)(35^{2} + 7)(53^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 39440

39440^{2} = (1^{2} + 4)(2^{2} + 4)(8^{2} + 4)(9^{2} + 4)(82^{2} + 4) = (1^{2} + 4)(2^{2} + 4)(76^{2} + 4)(82^{2} + 4)

= (1^{2} + 4)(8^{2} + 4)(26^{2} + 4)(82^{2} + 4) = (2^{2} + 4)(5^{2} + 4)(8^{2} + 4)(9^{2} + 4)(34^{2} + 4)

= (2^{2} + 4)(5^{2} + 4)(8^{2} + 4)(314^{2} + 4) = (2^{2} + 4)(5^{2} + 4)(34^{2} + 4)(76^{2} + 4)

= (5^{2} + 4)(8^{2} + 4)(26^{2} + 4)(34^{2} + 4) = (6^{2} + 4)(8^{2} + 4)(9^{2} + 4)(82^{2} + 4)

= (6^{2} + 4)(76^{2} + 4)(82^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 39442

39442^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39474

39474^{2}± 5 are primes.

39474^{2} = (1^{2} + 2)(5^{2} + 2)(10^{2} + 2)(16^{2} + 2)(27^{2} + 2) = (1^{2} + 2)(16^{2} + 2)(24^{2} + 2)(59^{2} + 2)

= (2^{2} + 2)(5^{2} + 2)(7^{2} + 2)(16^{2} + 2)(27^{2} + 2) = (2^{2} + 2)(10^{2} + 2)(27^{2} + 2)(59^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 39491

39491^{2} = 2029^{2} + 2031^{2} + 2033^{2} + 2035^{2} + ... + 2605^{2}.

by Yoshio Mimura, Kobe, Japan

## 39501

39501^{2} = (10^{2} - 1)(3970^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 39508

39508^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39512

39512^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39546

39546^{2} = (2^{2} + 9)(15^{2} + 9)(717^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 39560

39560^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39581

39581^{2} = 1566655561, a square with 3 kinds of digits 1,5,6.

by Yoshio Mimura, Kobe, Japan

## 39600

39600^{2} = (4^{2} - 1)(10^{2} - 1)(21^{2} - 1)(49^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 39601

39601 = 199^{2}, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 39606

39606^{2} = (3^{2} + 5)(6^{2} + 5)(8^{2} + 5)(199^{2} + 5) = (6^{2} + 5)(31^{2} + 5)(199^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 39610

39610^{2} = (5^{2} + 9)(34^{2} + 9)(199^{2} + 9) = (8^{2} + 4)(9^{2} + 4)(521^{2} + 4) = (76^{2} + 4)(521^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 39624

39624^{2} = (3^{2} + 3)(39^{2} + 3)(293^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 39634

39634^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39640

39640^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39650

39650^{2} = (1^{2} + 1)(5^{2} + 1)(8^{2} + 1)(682^{2} + 1) = (29^{2} + 4)(1364^{2} + 4)

= (1^{2} + 9)(4^{2} + 9)(28^{2} + 9)(89^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 39662

39662^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39666

39666^{2} = 1573391556, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 39690

39690^{2} = (1^{2} + 5)(2^{2} + 5)(4^{2} + 5)(10^{2} + 5)(115^{2} + 5) = (1^{2} + 5)(4^{2} + 5)(5^{2} + 5)(20^{2} + 5)(32^{2} + 5)

= (1^{2} + 5)(4^{2} + 5)(10^{2} + 5)(17^{2} + 5)(20^{2} + 5) = (1^{2} + 5)(20^{2} + 5)(25^{2} + 5)(32^{2} + 5)

= (2^{2} + 5)(3^{2} + 5)(5^{2} + 5)(20^{2} + 5)(32^{2} + 5) = (2^{2} + 5)(3^{2} + 5)(10^{2} + 5)(17^{2} + 5)(20^{2} + 5)

= (2^{2} + 5)(4^{2} + 5)(10^{2} + 5)(11^{2} + 5)(25^{2} + 5) = (2^{2} + 5)(4^{2} + 5)(25^{2} + 5)(115^{2} + 5)

= (2^{2} + 5)(5^{2} + 5)(7^{2} + 5)(10^{2} + 5)(32^{2} + 5) = (2^{2} + 5)(10^{2} + 5)(11^{2} + 5)(115^{2} + 5)

= (3^{2} + 5)(4^{2} + 5)(10^{2} + 5)(11^{2} + 5)(20^{2} + 5) = (3^{2} + 5)(4^{2} + 5)(20^{2} + 5)(115^{2} + 5)

= (4^{2} + 5)(7^{2} + 5)(10^{2} + 5)(115^{2} + 5) = (4^{2} + 5)(17^{2} + 5)(20^{2} + 5)(25^{2} + 5)

= (5^{2} + 5)(11^{2} + 5)(20^{2} + 5)(32^{2} + 5) = (10^{2} + 5)(11^{2} + 5)(17^{2} + 5)(20^{2} + 5)

= (17^{2} + 5)(20^{2} + 5)(115^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 39710

39710^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39717

39717^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 39732

39732^{2} = 897^{2} + 898^{2} + 899^{2} + ... + 1760^{2}.

by Yoshio Mimura, Kobe, Japan

## 39750

39750^{2} = (3^{2} + 6)(12^{2} + 6)(838^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 39776

39776^{2} = (2^{2} + 7)(85^{2} + 7)(141^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 39780

39780^{2} = (1^{2} + 9)(2^{2} + 9)(3^{2} + 9)(5^{2} + 9)(11^{2} + 9)(12^{2} + 9)

= (1^{2} + 9)(2^{2} + 9)(3^{2} + 9)(5^{2} + 9)(141^{2} + 9) = (1^{2} + 9)(2^{2} + 9)(5^{2} + 9)(9^{2} + 9)(63^{2} + 9)

= (1^{2} + 9)(2^{2} + 9)(5^{2} + 9)(15^{2} + 9)(39^{2} + 9) = (1^{2} + 9)(3^{2} + 9)(5^{2} + 9)(12^{2} + 9)(41^{2} + 9)

= (1^{2} + 9)(5^{2} + 9)(11^{2} + 9)(12^{2} + 9)(15^{2} + 9) = (1^{2} + 9)(5^{2} + 9)(15^{2} + 9)(141^{2} + 9)

= (2^{2} + 9)(3^{2} + 9)(5^{2} + 9)(11^{2} + 9)(39^{2} + 9) = (2^{2} + 9)(3^{2} + 9)(5^{2} + 9)(15^{2} + 9)(29^{2} + 9)

= (2^{2} - 1)(14^{2} - 1)(16^{2} - 1)(103^{2} - 1) = (3^{2} + 9)(5^{2} + 9)(11^{2} + 9)(141^{2} + 9)

= (3^{2} + 9)(5^{2} + 9)(39^{2} + 9)(41^{2} + 9) = (5^{2} + 9)(9^{2} + 9)(11^{2} + 9)(63^{2} + 9)

= (5^{2} + 9)(11^{2} + 9)(15^{2} + 9)(39^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 39785

39785^{2} = (8^{2} + 9)(10^{2} + 9)(446^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 39798

39798^{2} = (2^{2} + 2)(3^{2} + 2)(5^{2} + 2)(20^{2} + 2)(47^{2} + 2) = (5^{2} + 2)(8^{2} + 2)(20^{2} + 2)(47^{2} + 2)

= (5^{2} + 2)(20^{2} + 2)(382^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 39822

39822^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 39845

39845^{2} = 1587624025, with 15876^{2} = 126 and 24025 = 155^{2}.

by Yoshio Mimura, Kobe, Japan

## 39894

39894^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 39920

39920^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 39925

39925^{2} = (11^{2} + 4)(3571^{2} + 4).

by Yoshio Mimura, Kobe, Japan