## 43038

43038^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 43065

43065^{2} = 232^{3} + 233^{3} + 234^{3} + 235^{3} + 236^{3} + ... + 318^{3}.

by Yoshio Mimura, Kobe, Japan

## 43098

The sum of divisors of 43098^{2} is a cube 1729^{3}.

by Yoshio Mimura, Kobe, Japan

## 43134

43134^{2} = (12^{2} + 3)(32^{2} + 3)(111^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 43170

43170^{2}= 340 x 341 + 341 x 342 + 342 x 343 + 343 x 344 + ... + 1778 x 1779.

by Yoshio Mimura, Kobe, Japan

## 43176

43176^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 43200

43200^{2} = (2^{2} - 1)(3^{2} - 1)(7^{2} - 1)(26^{2} - 1)(49^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(11^{2} - 1)(26^{2} - 1)(31^{2} - 1)

= (2^{2} - 1)(4^{2} - 1)(7^{2} - 1)(19^{2} - 1)(49^{2} - 1) = (2^{2} - 1)(4^{2} - 1)(11^{2} - 1)(19^{2} - 1)(31^{2} - 1)

= (3^{2} - 1)(4^{2} - 1)(5^{2} - 1)(26^{2} - 1)(31^{2} - 1) = (3^{2} - 1)(4^{2} - 1)(9^{2} - 1)(17^{2} - 1)(26^{2} - 1)

= (3^{2} - 1)(19^{2} - 1)(26^{2} - 1)(31^{2} - 1) = (5^{2} - 1)(7^{2} - 1)(26^{2} - 1)(49^{2} - 1)

= (5^{2} - 1)(11^{2} - 1)(26^{2} - 1)(31^{2} - 1) = (9^{2} - 1)(11^{2} - 1)(17^{2} - 1)(26^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 43230

43230^{2} = (3^{2} + 6)(4^{2} + 6)(16^{2} + 6)(147^{2} + 6) = (16^{2} + 6)(18^{2} + 6)(147^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 43254

43254^{2} = (1^{2} + 2)(2^{2} + 2)(5^{2} + 2)(40^{2} + 2)(49^{2} + 2) = (1^{2} + 2)(5^{2} + 2)(22^{2} + 2)(218^{2} + 2)

= (4^{2} + 2)(5^{2} + 2)(40^{2} + 2)(49^{2} + 2) = (22^{2} + 2)(40^{2} + 2)(49^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 43273

43273^{2} = 1122^{2} + 1123^{2} + 1124^{2} + 1125^{2} + ... + 1915^{2}.

by Yoshio Mimura, Kobe, Japan

## 43290

43290^{2} = (19^{2} + 9)(24^{2} + 9)(93^{2} + 9) = (2^{2} + 9)(6^{2} + 9)(19^{2} + 9)(93^{2} + 9)

= (2^{2} + 9)(93^{2} + 9)(129^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 43296

43296^{2} = (4^{2} + 8)(6^{2} + 8)(22^{2} + 8)(60^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 43309

43309^{2} = 216^{3} + 217^{3} + 218^{3} + 219^{3} + 220^{3} + ... + 313^{3}.

by Yoshio Mimura, Kobe, Japan

## 43320

247^{2} + 43320 = 323^{2}, 247^{2} - 43320 = 133^{2}.

by Yoshio Mimura, Kobe, Japan

## 43321

43321843321 = 208139^{2}.

by Yoshio Mimura, Kobe, Japan

## 43362

43362^{2} = (2^{2} + 2)(3^{2} + 2)(5^{2} + 2)(12^{2} + 2)(85^{2} + 2) = (5^{2} + 2)(8^{2} + 2)(12^{2} + 2)(85^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 43382

43382^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 43413

43413^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 43428

43428^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 43470

43470^{2} = (1^{2} + 5)(2^{2} + 5)(4^{2} + 5)(8^{2} + 5)(10^{2} + 5)(15^{2} + 5)

= (1^{2} + 5)(2^{2} + 5)(5^{2} + 5)(8^{2} + 5)(130^{2} + 5) = (1^{2} + 5)(2^{2} + 5)(10^{2} + 5)(15^{2} + 5)(38^{2} + 5)

= (1^{2} + 5)(5^{2} + 5)(8^{2} + 5)(10^{2} + 5)(38^{2} + 5) = (2^{2} + 5)(4^{2} + 5)(5^{2} + 5)(15^{2} + 5)(38^{2} + 5)

= (2^{2} + 5)(4^{2} + 5)(8^{2} + 5)(15^{2} + 5)(25^{2} + 5) = (2^{2} + 5)(5^{2} + 5)(7^{2} + 5)(360^{2} + 5)

= (2^{2} + 5)(5^{2} + 5)(8^{2} + 5)(10^{2} + 5)(31^{2} + 5) = (2^{2} + 5)(7^{2} + 5)(15^{2} + 5)(130^{2} + 5)

= (2^{2} + 5)(8^{2} + 5)(10^{2} + 5)(11^{2} + 5)(15^{2} + 5) = (2^{2} + 5)(8^{2} + 5)(15^{2} + 5)(115^{2} + 5)

= (2^{2} + 5)(15^{2} + 5)(25^{2} + 5)(38^{2} + 5) = (3^{2} + 5)(4^{2} + 5)(8^{2} + 5)(15^{2} + 5)(20^{2} + 5)

= (3^{2} + 5)(15^{2} + 5)(20^{2} + 5)(38^{2} + 5) = (4^{2} + 5)(5^{2} + 5)(10^{2} + 5)(169^{2} + 5)

= (4^{2} + 5)(7^{2} + 5)(8^{2} + 5)(10^{2} + 5)(15^{2} + 5) = (4^{2} + 5)(10^{2} + 5)(15^{2} + 5)(61^{2} + 5)

= (4^{2} + 5)(15^{2} + 5)(20^{2} + 5)(31^{2} + 5) = (5^{2} + 5)(7^{2} + 5)(8^{2} + 5)(130^{2} + 5)

= (5^{2} + 5)(8^{2} + 5)(25^{2} + 5)(38^{2} + 5) = (5^{2} + 5)(61^{2} + 5)(130^{2} + 5)

= (7^{2} + 5)(10^{2} + 5)(15^{2} + 5)(38^{2} + 5) = (8^{2} + 5)(15^{2} + 5)(17^{2} + 5)(20^{2} + 5)

= (10^{2} + 5)(25^{2} + 5)(169^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 43484

43484^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 43512

43512^{2} = (1^{2} + 3)(5^{2} + 3)(12^{2} + 3)(16^{2} + 3)(21^{2} + 3) = (2^{2} + 3)(5^{2} + 3)(9^{2} + 3)(16^{2} + 3)(21^{2} + 3)

= (3^{2} + 3)(16^{2} + 3)(21^{2} + 3)(37^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 43533

43533^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 43630

43630^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 43646

43646^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 43650

43650^{2} = (1^{2} + 9)(6^{2} + 9)(31^{2} + 9)(66^{2} + 9) = (3^{2} + 9)(4^{2} + 9)(31^{2} + 9)(66^{2} + 9)

= (21^{2} + 9)(31^{2} + 9)(66^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 43680

394^{2} + 43680 = 446^{2}, 394^{2} - 43680 = 334^{2},

233^{2} + 43680 = 313^{2}, 233^{2} - 43680 = 103^{2},

218^{2} + 43680 = 302^{2}, 218^{2} - 43680 = 62^{2}.

43680^{2} = (3^{2} - 1)(4^{2} - 1)(6^{2} - 1)(25^{2} - 1)(27^{2} - 1) = (3^{2} - 1)(6^{2} - 1)(7^{2} - 1)(14^{2} - 1)(27^{2} - 1)

= (3^{2} - 1)(9^{2} - 1)(27^{2} - 1)(64^{2} - 1) = (3^{2} - 1)(14^{2} - 1)(27^{2} - 1)(41^{2} - 1)

= (6^{2} - 1)(11^{2} - 1)(25^{2} - 1)(27^{2} - 1) = (9^{2} - 1)(13^{2} - 1)(14^{2} - 1)(27^{2} - 1)

= (9^{2} - 1)(27^{2} - 1)(181^{2} - 1) = (14^{2} - 1)(15^{2} - 1)(209^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 43681

43681 = 209^{2}, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 43696

43696^{2}= 2602 x 2603 + 2603 x 2604 + 2604 x 2605 + 2605 x 2606 + ... + 2857 x 2858.

by Yoshio Mimura, Kobe, Japan

## 43708

43708^{2} = (1^{2} + 3)(37^{2} + 3)(590^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 43760

43760^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 43768

43768^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 43773

43773^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 43744

43744^{2} = 1913537536, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 43795

43795^{2} = 1256^{2} + 1257^{2} + 1258^{2} + 1259^{2} + ... + 1977^{2}.

by Yoshio Mimura, Kobe, Japan

## 43810

43810^{2} = (3^{2} + 4)(41^{2} + 4)(296^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 43812

43812^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 43824

43824^{2} = (16^{2} + 8)(18^{2} + 8)(148^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 43848

43848^{2} = (28^{2} - 1)(1567^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 43875

43875^{2} = (4^{2} + 9)(6^{2} + 9)(24^{2} + 9)(54^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 43902

43902^{2} = 1927385604, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 43923

43923^{2} = 1929229929, a square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 43940

43940^{2} = (1^{2} + 1)(3^{2} + 1)(5^{2} + 1)(8^{2} + 1)(239^{2} + 1).

by Yoshio Mimura, Kobe, Japan

## 43965

43965^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 43993

43993^{2} = 117^{2} + 118^{2} + 119^{2} + 120^{2} + ... + 1797^{2}.

by Yoshio Mimura, Kobe, Japan

## 43995

43995^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan