4702
47022 = 12 + 172 + 372 + 672
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4703
47032 = 22 + 232 + 262 + 682
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4704
47042 = 562 + 2802
A cubic polynomial :
(X + 652)(X + 722)(X + 47042) = X3 + 47052X2 + 4563122X + 220147202.
702 + 4704 = 982, 702 - 4704 = 142.
Komachi equations:
47042 = 12 * 22 / 32 / 42 * 562 * 72 * 82 * 92 = 92 * 82 * 72 / 62 / 52 * 42 / 32 * 2102
= 982 / 72 * 62 / 52 * 42 / 32 * 2102.
by Yoshio Mimura, Kobe, Japan
4705
A cubic polynomials :
(X + 652)(X + 722)(X + 47042) = X3 + 47052X2 + 4563122X + 220147202,
(X + 5602)(X + 18632)(X + 42842) = X3 + 47052X2 + 83989082X + 44694115202.
by Yoshio Mimura, Kobe, Japan
4707
47072 = 442 + 2172 + 2282
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4709
47092 = 22174681, 2 * 2 * 1 * 7 + 4681 = 22 - 1 + 7 + 4681 = 4709.
47094 = 491716477451761,
and 42 + 92 + 12 + 72 + 12 + 642 + 72 + 72 + 42 + 52 + 172 + 62 + 12 = 4709,
47094 = 491716477451761,
and 42 + 92 + 172 + 12 + 642 + 72 + 72 + 42 + 52 + 12 + 72 + 62 + 12 = 4709.
by Yoshio Mimura, Kobe, Japan
4710
47102 = 782 + 972 + 2752
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4712
47122 = 602 + 1642 + 2602 = 382 + 422 + 542 + 542
The square root of 4712 is 68.644..., and 68 = 62 + 42 + 42.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4715
47152 = 1162 + 1932 + 2382
47152 = (122 + 123 + 124 + ... + 144)2 + (145 + 146 + 147 + ... + 167)2 + (168 + 169 + 170 + ... + 190)2 + ... + (145 + 146 + 147 + ... + 167)2.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4721
47212 = 252 + 402 + 402 + 642.
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4724
47242 = 302 + 1982 + 2442 = 502 + 2202 + 2262
1 / 4724 = 0.0002116... and 2116 = 462. (also 4725).
The square root of 4724 is 68.7313..., and 68 = 72 + 32 + 12 + 32.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4725
47252 = (42 - 1)(62 - 1)(82 - 1)(262 - 1).
Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4728
47282 = 772 + 1422 + 2672
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4732
47322 = (12 + 3)(192 + 3)(1242 + 3) = (12 + 3)(22 + 3)(72 + 3)(1242 + 3)
= (52 + 3)(72 + 3)(1242 + 3).
47322 = 262 + 522 + 522 + 522
Page of Squares : First Upload September 4, 2008 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4733
47332 = 1052 + 1982 + 2382
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4734
12 + 22 + 32 + 42 + 52 + 62 + ... + 47342 = 35375379135, the first 11-digit sum whose digits are odd (there are 3 such sums in all. See 5382, 6102).
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4735
47352 = 1402 + 1932 + 2322
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4737
The square root of 4737 is 68.82..., and 68 = 82 + 22.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4738
47382 = 22448644, a square with even digits.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4742
47422± 3 are primes.
47422 = 72 + 1412 + 2702.
Page of Squares : First Upload September 4, 2008 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
4743
47432 = 62 + 412 + 562 + 562
47432 = 22496049, and 22 * 4 * 9 * 6 + 0 * 4 - 9 = 4743.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4744
47442 = 402 + 1742 + 2582.
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4745
47452 = (22 + 9)(82 + 9)(1542 + 9).
Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4746
47462 = 92 + 832 + 2802.
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4747
1 / 4747 = 0.000210659, 212 + 02 + 652 + 92 = 4747,
1 / 4747 = 0.000210659, 212 + 0652 + 92 = 4747.
by Yoshio Mimura, Kobe, Japan
4750
47502 = 22562500, and 225 = 152, 62500 = 2502.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4752
47522± 5 are primes.
47522 = (12 + 8)(22 + 8)(162 + 8)(282 + 8) = (12 + 8)(22 + 8)(42 + 8)(52 + 8)(162 + 8)
= (12 + 8)(42 + 8)(52 + 8)(62 + 8)(82 + 8) = (12 + 8)(42 + 8)(82 + 8)(382 + 8)
= (12 + 8)(62 + 8)(82 + 8)(282 + 8) = (102 + 8)(162 + 8)(282 + 8)
= (22 + 8)(42 + 8)(162 + 8)(172 + 8) = (22 + 8)(42 + 8)(2802 + 8) = (22 + 8)(52 + 8)(82 + 8)(282 + 8) = (42 + 8)(52 + 8)(102 + 8)(162 + 8) = (42 + 8)(62 + 8)(82 + 8)(172 + 8) = (62 + 8)(162 + 8)(442 + 8).
47522 = 192 + 532 + 2822 = 1082 + 1862 + 2462 = 1112 + 1692 + 2542
47522 = 22581504 appears in the decimal expressions of e:
e = 2.71828•••22581504••• (from the 29097th digit)
(22581504 is the fifth 8-digit square in the expression of e.)
by Yoshio Mimura, Kobe, Japan
4753
47532 = 492 + 1542 + 2662 = 12 + 342 + 462 + 642 = 282 + 422 + 422 + 632
47532 = 22591009, and 22 + 22 + 52 + 92 + 12 + 02 + 02 + 92 = 142.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4756
47562± 3 are primes.
47562 = 22619536, and 22 + 22 + 62 + 12 + 92 + 52 + 32 + 62 = 142.
Loop of length 4 :
4756 --- 472 + 562 = 5345 --- 532 + 452 = 4834 --- 482 + 342 = 3460 --- 342 + 602 = 4756
by Yoshio Mimura, Kobe, Japan
4759
47592 = 202 + 202 + 462 + 652
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4760
47602= 431 x 432 + 432 x 433 + 433 x 434 + 434 x 435 + ... + 528 x 529.
Komachi equation: 47602 = 982 * 7652 * 42 / 32 / 212.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 15, 2010by Yoshio Mimura, Kobe, Japan
4761
The square of 69.
47612 = 842 + 1202 + 2732
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4762
The square root of 4762 is 69.00724..., and 69 = 02 + 02 + 72 + 22 + 42.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4763
47635 = 2451336362939320043 : 22 + 452 + 132 + 32 + 62 + 362 + 22 + 92 + 32 + 92 + 322 + 02 + 02 + 42 + 32 = 22 + 452 + 132 + 362 + 32 + 62 + 22 + 92 + 32 + 92 + 322 + 02 + 02 + 42 + 32 = 4763.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4765
A cubic polynomial :
(X + 322)(X + 22052)(X + 42242) = X3 + 47652X2 + 93151682X + 2980454402.
by Yoshio Mimura, Kobe, Japan
4766
47662 = 22714756, and 2 + 2 + 7 - 1 + 4756 = 2 * 2 + 7 - 1 + 4756 = 4766.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4767
47672 = 22 + 92 + 342 + 682
47672 = (150 + 151 + 152)2 + (153 + 154 + 155)2 + ... + (294 + 295 + 296)2.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4768
Loop of length 10 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4768 - 6833 - 5713 - 3418 - ... - 2838 - 2228 - 1268 - 4768
(Note f(4768) = 472 + 682 = 6833, f(6833) = 682 + 332 = 5713, etc. See 1268)
by Yoshio Mimura, Kobe, Japan
4770
47702 = (32 + 9)(162 + 9)(692 + 9).
Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4771
The square root of 4771 is 69.0724..., and 69 = 02 + 72 + 22 + 42.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4772
47725 = 2474583854273733632 : 22 + 42 + 72 + 452 + 82 + 32 + 82 + 52 + 42 + 22 + 72 + 372 + 32 + 32 + 62 + 322 = 22 + 42 + 72 + 42 + 52 + 82 + 32 + 82 + 52 + 422 + 72 + 372 + 32 + 362 + 32 + 22 = 4772.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4775
47752 = 22800625, 2 - 2 + 800 * 6 - 25 = 2 / 2 * 800 * 6 - 25 = 4775.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4776
47762 = 682 + 1602 + 2642
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4779
47792 = (113 + 114 + 115)2 + (116 + 117 + 118)2 + (119 + 120 + 121)2 + ... + (287 + 288 + 289)2.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4780
47802 = 102 + 2182 + 2322
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4782
47825 = 2500620912904734432 : 22 + 52 + 02 + 02 + 62 + 202 + 92 + 12 + 292 + 02 + 472 + 342 + 42 + 32 + 22 = 4782.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4783
47832 = 22 + 182 + 182 + 692
Page of Squares : First Upload September 4, 2008 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4784
47845 = 2505854525373415424 : 22 + 52 + 02 + 52 + 82 + 52 + 452 + 22 + 52 + 372 + 342 + 12 + 52 + 42 + 22 + 42 = 4784.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
4785
47852 = (92 + 6)(5132 + 6).
47854 = 524237119250625, and 52 + 22 + 42 + 22 + 372 + 112 + 92 + 22 + 502 + 62 + 252 = 4785.
4785k + 12265k + 14355k + 16995k are squares for k = 1,2,3 (2202, 258502, 31339002).
Page of Squares : First Upload December 1, 2008 ; Last Revised December 28, 2013by Yoshio Mimura, Kobe, Japan
4788
47882 = 722 + 1732 + 2592
47882 = 22924944, a square consisting of just 3 kinds of digits.
Komachi equation: 47882 = 92 / 82 * 762 / 52 * 42 / 32 * 2102.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 15, 2010by Yoshio Mimura, Kobe, Japan
4789
47892 = 582 + 822 + 2812
47892 = 22934521, and 22 + 22 + 92 + 32 + 42 + 52 + 22 + 12 = 122.
Page of Squares : First Upload October 1, 2007 ; Last Revised September 4, 2008by Yoshio Mimura, Kobe, Japan
4795
192 + 432 + 672 + 912 + 1152 + ... + (24n + 19)2 + ... + 11712 = 47952.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4796
47962± 3 are primes.
47962 = 272 + 1202 + 2772
S2(4796) = S2(3531) + S2(4047), where S2(n) = 12 + 22 + 32 + ... + n2.
47962 = 23001616, and 2 + 300 * 16 - 1 * 6 = 2 + 300 * 16 * 1 - 6 = 4796.
Page of Squares : First Upload October 1, 2007 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
4797
47972 = 23011209, and 22 + 32 + 02 + 12 + 12 + 22 + 02 + 92 = 102.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4798
47982 = 23020804, a zigzag square.
47982 = 23020804, and 2 + 30 * 20 * 8 + 0 - 4 = 4798.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan
4799
47992 = 23030401, a zigzag square.
The square root of 4799 is 69.274..., and 69 = 22 + 72 + 42.
Page of Squares : First Upload October 1, 2007 ; Last Revised October 1, 2007by Yoshio Mimura, Kobe, Japan