## 34000

34000^{2} = (1^{2} + 4)(2^{2} + 4)(6^{2} + 4)(8^{2} + 4)(9^{2} + 4)(11^{2} + 4)

= (1^{2} + 4)(2^{2} + 4)(6^{2} + 4)(11^{2} + 4)(76^{2} + 4) = (1^{2} + 4)(4^{2} + 4)(9^{2} + 4)(14^{2} + 4)(26^{2} + 4)

= (1^{2} + 4)(6^{2} + 4)(8^{2} + 4)(11^{2} + 4)(26^{2} + 4) = (2^{2} + 4)(4^{2} + 4)(9^{2} + 4)(11^{2} + 4)(26^{2} + 4)

= (2^{2} + 4)(8^{2} + 4)(9^{2} + 4)(11^{2} + 4)(14^{2} + 4) = (2^{2} + 4)(11^{2} + 4)(14^{2} + 4)(76^{2} + 4)

= (8^{2} + 4)(11^{2} + 4)(14^{2} + 4)(26^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 34010

34010^{2}= 70 x 71 + 71 x 72 + 72 x 73 + ... + 1513 x 1514.

by Yoshio Mimura, Kobe, Japan

## 34020

34020^{2} = (1^{2} + 5)(2^{2} + 5)(3^{2} + 5)(5^{2} + 5)(11^{2} + 5)(20^{2} + 5)

= (1^{2} + 5)(2^{2} + 5)(4^{2} + 5)(5^{2} + 5)(7^{2} + 5)(25^{2} + 5)

= (1^{2} + 5)(2^{2} + 5)(5^{2} + 5)(7^{2} + 5)(10^{2} + 5)(11^{2} + 5) = (1^{2} + 5)(2^{2} + 5)(5^{2} + 5)(7^{2} + 5)(115^{2} + 5)

= (1^{2} + 5)(3^{2} + 5)(4^{2} + 5)(5^{2} + 5)(7^{2} + 5)(20^{2} + 5) = (1^{2} + 5)(3^{2} + 5)(7^{2} + 5)(20^{2} + 5)(25^{2} + 5)

= (3^{2} + 5)(5^{2} + 5)(7^{2} + 5)(11^{2} + 5)(20^{2} + 5) = (2^{2} + 5)(5^{2} + 5)(7^{2} + 5)(11^{2} + 5)(25^{2} + 5)

= (1^{2} + 5)(5^{2} + 5)(7^{2} + 5)(17^{2} + 5)(20^{2} + 5) = (1^{2} + 5)(5^{2} + 5)(25^{2} + 5)(101^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 34034

34034^{2} = 379^{2} + 380^{2} + 381^{2} + ... + 1522^{2}.

by Yoshio Mimura, Kobe, Japan

## 34048

34048^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34056

34056^{2} = (1^{2} + 8)(4^{2} + 8)(11^{2} + 8)(204^{2} + 8) = (4^{2} + 8)(28^{2} + 8)(247^{2} + 8)

= (5^{2} + 8)(11^{2} + 8)(16^{2} + 8)(32^{2} + 8) = (8^{2} + 8)(16^{2} + 8)(247^{2} + 8) = (16^{2} + 8)(2096^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 34071

34071^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 34076

34076^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34080

34080^{2}= 4009 x 4010 + 4010 x 4011 + 4011 x 4012 + ... + 4079 x 4080.

by Yoshio Mimura, Kobe, Japan

## 34099

34099^{2} = 329^{2} + 331^{2} + 333^{2} + 335^{2} + ... + 1913^{2}.

by Yoshio Mimura, Kobe, Japan

## 34100

34100^{2} = (2^{2} + 6)(4^{2} + 6)(26^{2} + 6)(88^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 34125

34125^{2} = 246^{2} + 247^{2} + 248^{2} + ... + 1519^{2}.

by Yoshio Mimura, Kobe, Japan

## 34188

34188^{2} = 351^{2} + 353^{2} + 355^{2} + 357^{2} + ... + 1917^{2}.

by Yoshio Mimura, Kobe, Japan

## 34203

34203^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 34228

34228^{2} = (13^{2} + 3)(14^{2} + 3)(185^{2} + 3).

(34228^{2} - 8) = (3^{2} - 8)(4^{2} - 8)(7^{2} - 8)(9^{2} - 8)(11^{2} - 8)(21^{2} - 8).

by Yoshio Mimura, Kobe, Japan

## 34272

34272^{2} = (3^{2} - 1)(7^{2} - 1)(35^{2} - 1)(50^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 34314

34314^{2} = (2^{2} + 3)(4^{2} + 3)(30^{2} + 3)(99^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 34320

34320^{2} = (2^{2} - 1)(12^{2} - 1)(21^{2} - 1)(79^{2} - 1) = (9^{2} - 1)(12^{2} - 1)(14^{2} - 1)(23^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 34336

34336^{2} = (1^{2} + 7)(15^{2} + 7)(797^{2} + 7) = (15^{2} + 7)(17^{2} + 7)(131^{2} + 7) = (43^{2} + 7)(797^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 34370

34370^{2} = (1^{2} + 6)(2^{2} + 6)(4108^{2} + 6) = (8^{2} + 6)(4108^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 34430

34430^{2} = 1788^{2} + 1789^{2} + 1790^{2} + ... + 2100^{2}.

by Yoshio Mimura, Kobe, Japan

## 34440

34440^{2} = (2^{2} - 1)(6^{2} - 1)(3361^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 34450

34450^{2} = (18^{2} + 1)(23^{2} + 1)(83^{2} + 1) = (2^{2} + 1)(8^{2} + 1)(23^{2} + 1)(83^{2} + 1)

= (1^{2} + 9)(2^{2} + 9)(4^{2} + 9)(16^{2} + 9)(37^{2} + 9) = (4^{2} + 9)(11^{2} + 9)(16^{2} + 9)(37^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 34461

34461^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 34504

34504^{2}= 967 x 968 + 968 x 969 + 969 x 970 + ... + 1647 x 1648.

by Yoshio Mimura, Kobe, Japan

## 34518

34518^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 34520

34520^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34534

34534^{2} = (1^{2})(2^{2})(3^{2} + 4^{3} + 5^{3} + ... + 963^{2}).

by Yoshio Mimura, Kobe, Japan

## 34560

204^{2} + 34560 = 276^{2}, 204^{2} - 34560 = 84^{2}.

34560^{2} = (2^{2} - 1)(3^{2} - 1)(5^{2} - 1)(7^{2} - 1)(11^{2} - 1)(19^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(5^{2} - 1)(9^{2} - 1)(161^{2} - 1)

= (2^{2} - 1)(4^{2} - 1)(5^{2} - 1)(7^{2} - 1)(9^{2} - 1)(17^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(7^{2} - 1)(19^{2} - 1)(31^{2} - 1)

= (2^{2} - 1)(7^{2} - 1)(9^{2} - 1)(17^{2} - 1)(19^{2} - 1) = (3^{2} - 1)(7^{2} - 1)(11^{2} - 1)(161^{2} - 1)

= (7^{2} - 1)(31^{2} - 1)(161^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 34578

34578^{2} = (1^{2} + 2)(2^{2} + 2)(26^{2} + 2)(313^{2} + 2) = (4^{2} + 2)(26^{2} + 2)(313^{2} + 2).

by Yoshio Mimura, Kobe, Japan

## 34596

34596 = 186^{2} is a square with different digits.

34596^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 34600

34600^{2} = (1^{2} + 4)(6^{2} + 4)(13^{2} + 4)(186^{2} + 4) = (13^{2} + 4)(14^{2} + 4)(186^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 34608

34608^{2} = (1^{2} + 3)(3^{2} + 3)(5^{2} + 3)(10^{2} + 3)(93^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 34634

34634^{2} = 1199513956, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 34641

34641^{2} = 1199998881 is a square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 34642

34642^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34650

34650^{2} = (1^{2} + 6)(3^{2} + 6)(12^{2} + 6)(15^{2} + 6)(18^{2} + 6) = (3^{2} + 6)(6^{2} + 6)(7^{2} + 6)(12^{2} + 6)(15^{2} + 6)

= (3^{2} + 6)(12^{2} + 6)(15^{2} + 6)(48^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 34656

190^{2} + 34656 = 266^{2}, 190^{2} - 34656 = 38^{2}.

by Yoshio Mimura, Kobe, Japan

## 34670

34670^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34680

221^{2} + 34680 = 289^{2}, 221^{2} - 34680 = 119^{2}.

34^{3} + 34680 = 272^{2}, 34^{3} - 34680 = 68^{2}.

by Yoshio Mimura, Kobe, Japan

## 34686

34686^{2} = (1^{2} + 5)(6^{2} + 5)(29^{2} + 5)(76^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 34710

34710^{2} = (2^{2} + 9)(9^{2} + 9)(13^{2} + 9)(76^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 34776

34776^{2} = (2^{2} - 1)(3^{2} - 1)(22^{2} - 1)(323^{2} - 1) = (5^{2} - 1)(22^{2} - 1)(323^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 34784

34784^{2} = (5^{2} + 7)(6149^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 34827

34827^{2} = 1212919929 is a square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 34830

34830^{2} = (5^{2} + 5)(9^{2} + 5)(20^{2} + 5)(34^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 34840

34840^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34848

34848^{2} = (4^{2} + 8)(6^{2} + 8)(28^{2} + 8)(38^{2} + 8) = (6^{2} + 8)(8^{2} + 8)(16^{2} + 8)(38^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 34850

34850^{2} = (1^{2} + 1)(2^{2} + 1)(4^{2} + 1)(7^{2} + 1)(378^{2} + 1) = (1^{2} + 1)(2^{2} + 1)(9^{2} + 1)(32^{2} + 1)(38^{2} + 1)

= (2^{2} + 1)(3^{2} + 1)(13^{2} + 1)(378^{2} + 1) = (2^{2} + 1)(3^{2} + 1)(4^{2} + 1)(9^{2} + 1)(132^{2} + 1)

= (2^{2} + 1)(4^{2} + 1)(9^{2} + 1)(13^{2} + 1)(32^{2} + 1) = (2^{2} + 1)(9^{2} + 1)(13^{2} + 1)(132^{2} + 1)

= (3^{2} + 1)(4^{2} + 1)(7^{2} + 1)(378^{2} + 1) = (3^{2} + 1)(9^{2} + 1)(32^{2} + 1)(38^{2} + 1)

= (4^{2} + 1)(7^{2} + 1)(9^{2} + 1)(132^{2} + 1) = (7^{2} + 1)(13^{2} + 1)(378^{2} + 1)

= (1^{2} + 4)(59^{2} + 4)(264^{2} + 4) = (9^{2} + 4)(59^{2} + 4)(64^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 34856

34856^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34866

34866^{2} = (2^{2} + 9)(3^{2} + 9)(17^{2} + 9)(132^{2} + 9) = (15^{2} + 9)(17^{2} + 9)(132^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 34874

34874^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34881

34881^{2} = (15^{2} + 6)(2295^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 34884

34884^{2} = (1^{2} + 8)(2^{2} + 8)(3^{2} + 8)(26^{2} + 8)(31^{2} + 8)

= (1^{2} + 2)(2^{2} + 2)(4^{2} + 2)(6^{2} + 2)(7^{2} + 2)(44^{2} + 2)

= (1^{2} + 2)(2^{2} + 2)(4^{2} + 2)(6^{2} + 2)(13^{2} + 2)(24^{2} + 2) = (1^{2} + 8)(2^{2} + 8)(7^{2} + 8)(14^{2} + 8)(31^{2} + 8)

= (1^{2} + 8)(2^{2} + 8)(14^{2} + 8)(235^{2} + 8) = (1^{2} + 8)(3^{2} + 8)(7^{2} + 8)(14^{2} + 8)(26^{2} + 8)

= (1^{2} + 2)(4^{2} + 2)(6^{2} + 2)(24^{2} + 2)(32^{2} + 2) = (1^{2} + 8)(14^{2} + 8)(26^{2} + 8)(31^{2} + 8)

= (2^{2} + 8)(3^{2} + 8)(7^{2} + 8)(10^{2} + 8)(31^{2} + 8) = (2^{2} + 8)(3^{2} + 8)(10^{2} + 8)(235^{2} + 8)

= (2^{2} + 2)(5^{2} + 2)(6^{2} + 2)(10^{2} + 2)(44^{2} + 2) = (2^{2} + 2)(6^{2} + 2)(7^{2} + 2)(10^{2} + 2)(32^{2} + 2)

= (2^{2} + 2)(10^{2} + 2)(32^{2} + 2)(44^{2} + 2) = (3^{2} + 8)(10^{2} + 8)(26^{2} + 8)(31^{2} + 8)

= (7^{2} + 8)(10^{2} + 8)(14^{2} + 8)(31^{2} + 8) = (10^{2} + 8)(14^{2} + 8)(235^{2} + 8).

by Yoshio Mimura, Kobe, Japan

## 34892

34892^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34932

34932^{2} = (7^{2} - 1)(5042^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 34934

34934^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34944

34944^{2} = (7^{2} - 1)(15^{2} - 1)(337^{2} - 1).

340^{2} + 34944 = 388^{2}, 340^{2} - 34944 = 284^{2}.

by Yoshio Mimura, Kobe, Japan

## 34954

34954^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 34960

34960^{2}= 401 x 402 + 402 x 403 + 403 x 404 + ... + 1550 x 1551.

by Yoshio Mimura, Kobe, Japan

## 34962

34962^{2} = 1222341444, a square every digit of which is non-zero and smaller than 5.

by Yoshio Mimura, Kobe, Japan

## 34965

34965^{2} = 1222551225 is a square with 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 34974

34974^{2} = (13^{2} + 5)(14^{2} + 5)(187^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 34992

34992^{2} = 1927^{2} + 1929^{2} + 1931^{2} + 1933^{2} + ... + 2437^{2}.

by Yoshio Mimura, Kobe, Japan