16017
160172± 2 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16023
A cubic polynomial:
(X + 35842)(X + 38882)(X + 16023) = X3 + 168732X2 + 858658082X + 2232739676162.
by Yoshio Mimura, Kobe, Japan
16032
160322± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16038
160382± 5 are primes.
160382 = (12 + 2)(42 + 2)(52 + 2)(192 + 2)(222 + 2) = (52 + 2)(222 + 2)(1402 + 2).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16065
160652 = (82 - 1)(20242 - 1).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16080
160802 = (13 + 5)(103 + 5)(353 + 5).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16098
160982± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16106
161062± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16107
161072 = S2(375) + S2(898), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16128
161282 = (22 - 1)(32 - 1)(132 - 1)(152 - 1)(172 - 1) = (22 - 1)(32 - 1)(52 - 1)(72 - 1)(972 - 1)
= (22 - 1)(52 - 1)(152 - 1)(1272 - 1) = (32 - 1)(72 - 1)(152 - 1)(552 - 1)
= (52 - 1)(132 - 1)(152 - 1)(172 - 1) = (23 + 8)(43 + 8)(63 + 8)(103 + 8).
by Yoshio Mimura, Kobe, Japan
16136
161362± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16149
161492 = (42 + 5)(35242 + 5).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16150
161502 = (12)(22)(32 + 43 + 53 + ... + 5802).
Page of Squares : First Upload February 9, 2013 ; Last Revised February 9, 2013by Yoshio Mimura, Kobe, Japan
16151
161512 = S2(39) + S2(921), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16169
161692 = 2042 + 2052 + 2062 + ... + 9252.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16170
161702 = (12 + 6)(152 + 6)(182 + 6)(222 + 6) = (12 + 6)(22 + 6)(42 + 6)(152 + 6)(272 + 6)
= (12 + 6)(32 + 6)(42 + 6)(152 + 6)(222 + 6) = (12 + 6)(32 + 6)(62 + 6)(82 + 6)(292 + 6)
= (12 + 6)(42 + 6)(272 + 6)(482 + 6) = (12 + 6)(42 + 6)(62 + 6)(132 + 6)(152 + 6)
= (12 + 6)(42 + 6)(62 + 6)(72 + 6)(272 + 6) = (12 + 6)(62 + 6)(152 + 6)(622 + 6)
= (12 + 6)(62 + 6)(72 + 6)(82 + 6)(152 + 6) = (12 + 6)(82 + 6)(152 + 6)(482 + 6)
= (152 + 6)(222 + 6)(482 + 6) = (22 + 6)(62 + 6)(272 + 6)(292 + 6)
= (32 + 6)(42 + 6)(72 + 6)(1202 + 6) = (32 + 6)(62 + 6)(222 + 6)(292 + 6)
= (42 + 6)(82 + 6)(152 + 6)(272 + 6) = (62 + 6)(272 + 6)(922 + 6)
= (62 + 6)(72 + 6)(152 + 6)(222 + 6) = (72 + 6)(182 + 6)(1202 + 6).
by Yoshio Mimura, Kobe, Japan
16174
161742± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16192
161922 = (12 + 7)(22 + 7)(42 + 7)(132 + 7)(272 + 7) = (12 + 7)(272 + 7)(2112 + 7)
= (12 + 7)(42 + 7)(52 + 7)(2112 + 7) = (192 + 7)(272 + 7)(312 + 7)
= (22 + 7)(32 + 7)(42 + 7)(132 + 7)(192 + 7) = (22 + 7)(32 + 7)(42 + 7)(92 + 7)(272 + 7)
= (22 + 7)(42 + 7)(192 + 7)(532 + 7) = (22 + 7)(42 + 7)(52 + 7)(92 + 7)(192 + 7)
= (22 + 7)(52 + 7)(132 + 7)(652 + 7) = (22 + 7)(652 + 7)(752 + 7)
= (22 + 7)(92 + 7)(192 + 7)(272 + 7) = (32 + 7)(192 + 7)(2112 + 7)
= (32 + 7)(42 + 7)(272 + 7)(312 + 7) = (42 + 7)(52 + 7)(192 + 7)(312 + 7)
= (42 + 7)(92 + 7)(132 + 7)(272 + 7).
by Yoshio Mimura, Kobe, Japan
16198
16198161984 = 1272722.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16200
162002 = (22 - 1)(42 - 1)(52 - 1)(192 - 1)(262 - 1) = (42 - 1)(262 - 1)(1612 - 1).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16212
162122 = (12 + 3)(32 + 3)(23402 + 3).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16236
162362 = (12 + 8)(382 + 8)(1422 + 8) = (12 + 8)(52 + 8)(62 + 8)(1422 + 8)
= (192 + 8)(222 + 8)(382 + 8) = (22 + 2)(42 + 2)(302 + 2)(522 + 2)
= (52 + 8)(62 + 8)(192 + 8)(222 + 8) = (62 + 8)(172 + 8)(1422 + 8).
by Yoshio Mimura, Kobe, Japan
16248
162482± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16250
162502 = (22 + 1)(72 + 1)(182 + 1)(572 + 1)
= (22 + 9)(42 + 9)(112 + 9)(792 + 9) = (42 + 9)(412 + 9)(792 + 9).
by Yoshio Mimura, Kobe, Japan
16254
162542 = (12 + 5)(22 + 5)(42 + 5)(92 + 5)(522 + 5) = (12 + 5)(32 + 5)(342 + 5)(522 + 5)
= (22 + 5)(32 + 5)(42 + 5)(92 + 5)(342 + 5) = (22 + 5)(92 + 5)(112 + 5)(522 + 5)
= (22 + 5)(92 + 5)(172 + 5)(342 + 5) = (42 + 5)(72 + 5)(92 + 5)(522 + 5)
= (42 + 5)(92 + 5)(112 + 5)(342 + 5).
by Yoshio Mimura, Kobe, Japan
16262
162622 = 264452644.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16275
162752 = (32 + 6)(52 + 6)(132 + 6)(572 + 6).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16288
162882± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16296
162962 = (12 + 3)(22 + 3)(32 + 3)(52 + 3)(1682 + 3) = (12 + 3)(52 + 3)(92 + 3)(1682 + 3).
162962 = 3333 + 3343 + 3353 + ... + 3393.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16302
163022± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16320
163202 = (162 - 1)(312 - 1)(332 - 1) = (22 - 1)(32 - 1)(332 - 1)(1012 - 1)
= (32 - 1)(112 - 1)(162 - 1)(332 - 1) = (52 - 1)(332 - 1)(1012 - 1).
by Yoshio Mimura, Kobe, Japan
16332
163322 = 652 + 662 + 672 + ... + 9282.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16350
163502 = (12 + 9)(42 + 9)(102 + 9)(992 + 9).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16354
163542 = (42 + 1)(62 + 1)(212 + 1)(312 + 1).
Page of Squares : First Upload November 9, 2013 ; Last Revised November 9, 2013by Yoshio Mimura, Kobe, Japan
16372
163722± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16376
163762 = 197 * 198 + 198 * 199 + 199 * 200 + 200 * 201 + ... + 932 * 933.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16380
163802 = (22 - 1)(62 - 1)(252 - 1)(642 - 1) = (62 - 1)(82 - 1)(142 - 1)(252 - 1).
163802 = 783 + 793 + 803 + 813 + ... + 1823.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16384
16384 is a zigzag square (1282) with different digits.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16400
164002 = (12 + 4)(62 + 4)(182 + 4)(642 + 4) = (142 + 4)(182 + 4)(642 + 4).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16416
164162 = (12 + 8)(22 + 8)(42 + 8)(122 + 8)(262 + 8) = (22 + 8)(42 + 8)(72 + 8)(102 + 8)(122 + 8)
= (42 + 8)(102 + 8)(122 + 8)(262 + 8).
by Yoshio Mimura, Kobe, Japan
16432
164322± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16434
164342± 5 are primes.
164342 = (12 + 2)(32 + 2)(42 + 2)(92 + 2)(742 + 2) = (12 + 2)(92 + 2)(142 + 2)(742 + 2).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16443
164432 = (22 + 5)(42 + 5)(162 + 5)(742 + 5).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16445
164452 = 270438025, and 2704 = 522, 38025 = 1952.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16455
164552± 2 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16464
164642 = (12 + 3)(32 + 3)(52 + 3)(122 + 3)(372 + 3) = (22 + 3)(32 + 3)(52 + 3)(92 + 3)(372 + 3).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16484
164842 is the sum of (12m + 6)2 for m = 0,1,2,...,168.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16492
164922 = (12 + 3)(42 + 3)(232 + 3)(822 + 3).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16498
164982 = (12 + 1)(152 + 1)(7762 + 1).
Page of Squares : First Upload November 9, 2013 ; Last Revised November 9, 2013by Yoshio Mimura, Kobe, Japan
16520
165202± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16524
165242 = (12 + 8)(102 + 8)(142 + 8)(372 + 8) = (12 + 8)(102 + 8)(5302 + 8)
= (12 + 8)(22 + 8)(32 + 8)(102 + 8)(372 + 8) = (22 + 2)(42 + 2)(72 + 2)(102 + 2)(222 + 2).
by Yoshio Mimura, Kobe, Japan
16549
165492 = 273869401 is a square consisting of different digits.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16552
165522± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16560
165602 = (42 - 1)(472 - 1)(912 - 1).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16587
165872 = (72 + 8)(21972 + 8).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16590
165902 = (22 + 6)(32 + 6)(62 + 6)(2092 + 6) = (62 + 6)(122 + 6)(2092 + 6).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16623
166232± 2 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16632
166322 = (22 - 1)(32 - 1)(82 - 1)(102 - 1)(432 - 1) = (22 - 1)(52 - 1)(102 - 1)(1972 - 1)
= (52 - 1)(82 - 1)(102 - 1)(432 - 1).
by Yoshio Mimura, Kobe, Japan
16635
166352 = 23282 + 23292 + 23302 + ... + 23772.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16641
166412± 2 are primes.
16641 is a square (1292) with 3 kinds of digits.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16650
166502 = (32 + 9)(192 + 9)(2042 + 9) = (42 + 9)(92 + 9)(182 + 9)(192 + 9)
= (42 + 9)(92 + 9)(3512 + 9) = (62 + 9)(192 + 9)(1292 + 9).
by Yoshio Mimura, Kobe, Japan
16659
166592± 2 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16665
166652 = 277722225 is a square with 3 kinds of digits (2,5,7).
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16667
166672 = 277788889 is a square with non-decreasing digits.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16670
166702 = 15672 + 15692 + 15712 + 15732 + ... + 17652.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16676
166762 = (62 + 8)(25142 + 8).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16686
166862 = 2973 + 4683 + 5313.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16699
(166992 -7) = (32 - 7)(42 - 7)(102 - 7)(162 - 7)(262 - 7).
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16705
167052 = (82 + 1)(20722 + 1).
Page of Squares : First Upload November 9, 2013 ; Last Revised November 9, 2013by Yoshio Mimura, Kobe, Japan
16714
167142 = S2(621) + S2(842), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16737
167372± 2 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16758
167582± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16772
167722± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16780
The quadratic polynomial 16780X2 - 59500X + 99841 takes the values 2392, 2192, 2692, 3612, 4712, 5892 at X = 1, 2, ..., 6
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16800
168002 = (22 - 1)(32 - 1)(42 - 1)(92 - 1)(992 - 1) = (22 - 1)(92 - 1)(112 - 1)(992 - 1)
= (32 - 1)(42 - 1)(62 - 1)(92 - 1)(292 - 1) = (42 - 1)(52 - 1)(92 - 1)(992 - 1)
= (42 - 1)(62 - 1)(152 - 1)(492 - 1) = (62 - 1)(92 - 1)(112 - 1)(292 - 1) = (92 - 1)(192 - 1)(992 - 1).
by Yoshio Mimura, Kobe, Japan
16808
168082± 3 are primes.
168082 = (12 + 7)(392 + 7)(1522 + 7).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16817
168172 = 1972 + 1982 + 1992 + ... + 9492.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16834
168342± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16843
p = 16843 is a counter example to the conjecture:
2p-1Cp-1-1 is divisible by p2 iff p is prime.
(the next counter prime is greater than 150000).
by Yoshio Mimura, Kobe, Japan
16854
168542 = 284057316 is a square consisting of different digits.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16872
168722 = (152 + 3)(212 + 3)(532 + 3) = (32 + 3)(42 + 3)(212 + 3)(532 + 3).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16873
A cubic polynomial:
(X + 35842)(X + 38882)(X + 160232) = X3 + 168732X2 + 858658082X + 2232739676162.
by Yoshio Mimura, Kobe, Japan
16878
168782 = 284866884 is a square with even digits.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16884
168842± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16900
16900 = 1302, 16 = 42, 900 = 302.
169002 = (12 + 1)(22 + 1)(32 + 1)(72 + 1)(2392 + 1) = (12 + 4)(32 + 4)(42 + 4)(162 + 4)(292 + 4)
= (162 + 4)(292 + 4)(362 + 4) = (32 + 4)(42 + 4)(292 + 4)(362 + 4).
by Yoshio Mimura, Kobe, Japan
16905
169052 = (42 + 5)(102 + 5)(3602 + 5).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16915
169152 = 3952 + 3962 + 3972 + ... + 9722.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16918
169182± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16920
169202 = (42 - 1)(462 - 1)(952 - 1).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16926
169262 = (22 + 3)(62 + 3)(202 + 3)(512 + 3) = (62 + 3)(112 + 3)(122 + 3)(202 + 3)
= (62 + 3)(202 + 3)(1352 + 3) = (62 + 3)(332 + 3)(822 + 3).
by Yoshio Mimura, Kobe, Japan
16928
169282 = (12 + 7)(42 + 7)(192 + 7)(652 + 7).
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16932
169322 = (142 + 8)(182 + 8)(652 + 8) = (22 + 8)(32 + 8)(182 + 8)(652 + 8).
169322 = 848 * 849 + 849 * 850 + 850 * 851 851 * 852 + ... + 1136 * 1137.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16954
169542± 3 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16958
169582 = S2(598) + S2(865), where S2(n) = 12 + 22 + 32 + ... + n2.
Page of Squares : First Upload January 28, 2012 ; Last Revised January 28, 2012by Yoshio Mimura, Kobe, Japan
16962
169622± 5 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan
16983
169832± 2 are primes.
Page of Squares : First Upload January 25, 2014 ; Last Revised January 25, 2014by Yoshio Mimura, Kobe, Japan