A collection of sums of 2 squares, sums of 2 cubes, sums of 3 squares, sums of 3 cubes, and of other powers.
1 = 3^2 - 2^3
2 = 3^3 - 5^2
2 = 4^2 - 3^2 - 2^2 - 1^2
3 = 2^2 - 2^0
3^2 = 5^2 - 4^2
3^3 = 6^3 - 5^3 - 4^3
3^3 = 3^2 + 3^2 + 3^2
4 = 3^2 - 2^2 - 1^2
5 = 3^2 - 2^2 = 1^2 + 2^2
5 = 2^5 - 3^3 ............ 5^2 = 3^2 + 4^2 = 13^2 - 12^2
6 = 3 + 2 + 1 .......... (the smallest perfect number)
7 = 2^5 - 5^2 ................... (7 is quite remarkable, notice the digits)
7 = 4 + 3 = 4^2 - 3^2
7 = 1^2 + 1^2 + 1^2 + 2^2
8 = 2^4 - 2^3 ........ = 2^2 + 2^2
9 = 1^3 + 2^3
10 = 2 + 3 + 5 (sum of first prime numbers) .... 100 = 10^2 = (2 + 3 + 5)^2
11 = 6 + 5 = 6^2 - 5^2
12 = 3^1 + 3^2
12 = 2^4 - 2^2
12 = 4^2 - 4^1
13 = 2^2 + 3^2 = 7^2 - 6^2 .... 13^2 = 8^3 - 7^3
13 = 2^2 + 3^2 ........... 13^2 = 5^2 + 12^2 ... 13^4 = 120^2 + 121^2
14 = 1^2 + 2^2 + 3^2
14 = 1 + 4 + 9 = 1^2 + 2^2 + 3^2
14 = 2 + 3 + 4 + 5 (sum of consecutive integers)
14 = 2^1 + 2^2 + 2^3
15 is a triangular number: 15 = 1 + 2 + 3 + 4 + 5
15 = 8^2 - 7^2
15 = 4^2 - 1^2
16 = 1 + 3 + 5 + 7 (sum of the four first odd numbers)
17 = 2 + 3 + 5 + 7 (sum of 4 consecutive prime numbers)
17 = 2^3 + 3^2 = 1^3 + 4^2 .................. = 1^4 + 2^4
17 = = 9^2 - 8^2 = 3^4 - 4^3 ........ 17^2 = 1^3 + 2^3 + 4^3 + 6^3
17 = 2^(2^2) + 1 (third Fermat prime)
17 = 2 + 2 + 13 = 3 + 3 + 11 = 3 + 7 + 7 = 5 + 5 + 7
(17 is the smallest number with 4 representations as a sum of 3 primes)
18 = 3^3 - 3^2
18 = 3 + 4 + 5 + 6 (sum of consecutive numbers)
19 = 3^3 - 2^3 (difference of consecutive cubes)
20 = 6^2 - 4^2 ........... 20^2 = 7^0 + 7^1 + 7^2 + 7^3 = 29^2 - 21^2
20 = 1 + 3 + 6 + 10 (sum of the first 4 triangular numbers)
20 = 1 + 1 + 2 + 3 + 5 + 8 (sum of the first 6 Fibonacci numbers)
20 = 2 + 4 + 6 + 8 (sum of the first 4 even numbers)
21 = 4^0 + 4^1 + 4^2
22 = 4 + 5 + 6 + 7
22 = 1^4 + 2^3 + 3^2 + 4^1
23 = 5 + 7 + 11 (smallest prime number that is a sum of 3 consecutive prime numbers)
23 = 12^2 - 11^2
23 = 1^4 + 2^3 + 3^2 + 4^1 + 5^0
24 = 3 + 5 + 7 + 9 (sum of consecutive odd numbers)
24 = 2^5 - 2^3
24 = 3^3 - 3^1
25 = 1^2 + 2^2 + 2^2 + 4^2
25 = 4^2 + 3^2 ........... 25^2 = 7^2 + 24^2
26 = 5 + 8 + 13 (sum of consecutive Fibonacci numbers)
27 (=3^3) = 3^2 + 3^2 + 3^2
27 = 1^2 + 1^2 + 5^2 ............... 19683 = 27^3 = 3^3 + 18^3 + 24^3
27 = 3^2 + 3^2 + 3^2
27 = 6^2 - 3^2
27 = 14^2 - 13^2 (difference of consecutive squares)
27 = 2 + 3 + 4 + 5 + 6 + 7 (sum of consecutive natural numbers)
28 = 1 + 5 + 9 + 13 (hexagonal number)
28 = 2 + 3 + 5 + 7 + 11 (sum of the first five consecutive primes)
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 (sum of the first 7 consecutive numbers)
28 = 2^5 – 2^2
29 is the smallest prime of the form 7n + 1 ... 29^2 = 21^2 + 20^2 (Pythagorean triple)
29 = 2^2 + 3^2 + 4^2 (sum of 3 consecutive squares)
29 = 3 + 5 + 7 + 11 + 13 (sum of consecutive primes)
29 = 2^2 + 3^2 + 4^2
30 = 1^2 + 2^2 + 3^2 + 4^2
30 = 9 + 10 + 11 = 6 + 7 + 8 + 9 = 4 + 5 + 6 + 7 + 8
31 = 2^2 + 3^3 ............................... ..... = 2^5 - 1
31 = 5^0 + 5^1 + 5^2
31 = 2^0 + 2^1 + 2^2 + 2^3 + 2^4
32:
33:
34:
35:
36 = 2^2 + 4^2 + 4^2
36 (=6^2) = 1^3 + 2^3 + 3^3
36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19
(smallest number with four representations as a sum of two distinct prime numbers)
37:
38:
39:
40:
41:
42:
43:
44:
45:
46:
47:
48 = 4^2 + 4^2 + 4^2
48 = 3 + 5 + 7 + 9 + 11 + 13
48 = 5 + 43 = 7 + 41 = 11 + 37 = 17 + 31 = 19 + 29
(the smallest number with five representations as a sum of two primes)
49:
50 = 1^2 + 7^2 = 5^2 + 5^2 = 3^2 + 4^2 + 5^2
51 = 2^3 + 2^3 + 2^3 + 3^3
52 = 5^2 + 3^3
53 = 2^2 + 7^2 = 1^2 + 4^2 + 6^2 ......... 53^2 = 28^2 + 45^2
54 = 3^3 + 3^3
54 = 7^2 + 2^2 + 1^2 = 6^2 + 3^2 + 3^2 = 5^2 + 5^2 + 2^2
(54 is the smallest number that can be written as the sum of 3 squares in 3 different ways)
55 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2
56:
57 = 1^2 + 2^2 + 4^2 + 6^2 ................................. = 7 + 8 + 9 + 10 + 11 + 12
58 = 3^2 + 7^2 ....... = 2^2 + 2^2 + 5^2 + 5^2
59 = 2^3 + 2^3 + 2^3 + 2^3 + 3^3
60:
61 = 5^2 + 6^2 ............................ 61^2 = 11^2 + 60^2
62 = 2^3 + 3^3 + 3^3
62 = 1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2
(62 is the smallest number that can be written as the sum of 3 distinct squares in 2 distinct ways)
63 = 6^2 + 3^3
64:
65 = 1^2 + 8^2 = 4^2 + 7^2 ........... = 1^5 + 2^4 + 3^3 + 4^2 + 5^1
66 = 2^2 + 2^2 + 3^2 + 7^2
67:
68:
69:
70 = 3^2 + 5^2 + 6^2 ........... 70^2 = 1^2 + 2^2 + 3^2 + 4^2 + ... + 22^2 + 23^2 + 24^2
71 -> 71^2 = 2^7 + 17^3 (sum of prime powers of two prime numbers)
71 -> 71^4 = 136^3 + 4785^2
72 = 2^3 + 4^3 ..... 72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
73 = 1^3 + 2^3 + 4^3
73 = 3^2 + 8^2 ........................... 73^2 = 48^2 + 55^2 (Pythagorean triple)
74:
75 -> 75^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
76 = 2^2 + 6^2 + 6^2 .............................. = 3^2 + 3^2 + 3^2 + 7^2
77 = 4^2 + 5^2 + 6^2
78 = 2^2 + 5^2 + 7^2 ............ 78^3 = 39^3 + 52^3 + 65^3
79 = 2^2 + 5^2 + 5^2 + 5^2
80 = 4^2 + 8^2 = 4^2 + 4^3 ........ 80^2 = 4^3 + 8^3 + 12^3 + 16^3
80 = 2^3 + 2^3 + 4^3
81 = 1^2 + 4^2 + 8^2 ......... 2^5 + 7^2
82 = 1^2 + 9^2 .............. 2^2 + 2^2 + 5^2 + 7^2
83:
84 = 4^1 + 4^2 + 4^3
84 = 3^2 + 5^2 + 5^2 + 5^2
84 = 5 + 79 = 11 + 73 = 13 + 71 = 17 + 67 = 23 + 61 = 31 + 53 = 37 + 47 = 41 + 43
(smallest number with eight representations as a sum of two primes)
84 = 2^2 + 4^2 + 8^2
85 = 2^2 + 9^2 ........................... 85^2 = 13^2 + 84^2 = 36^2 + 77^2
85 = 6^2 + 7^2 .......... 4^0 + 4^1 + 4^2 + 4^3 (sum of powers of 4)
86 = 3^2 + 4^2 + 5^2 + 6^2 (sum of consecutive squares)
87 = 2^2 + 3^2 + 5^2 + 7^2
88:
89 = 8^1 + 9^2 ........... 89^2 = 39^2 + 80^2 (Pythagorean triple)
89 = 2^3 + 3^3 + 3^3 + 3^3
90 = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 (sum of consecutive squares)
90 = 9^1 + 9^2 = 10^2 - 10^1
91 = 3^3 + 4^3
92:
93 = 2^2 + 5^2 + 8^2
94:
95:
96 = 4^2 + 4^2 + 8^2
97:
98:
99 = 2^3 + 3^3 + 4^3
100 = 6^2 + 8^2 ....... = 1^3 + 2^3 + 3^3 + 4^3
101
102
103
104
105
106
107
108
109 = 3^2 + 10^2 ........... 109^2 = 60^2 + 91^2
110:
111
112
113
114
115 = 3^2 + 5^2 + 9^2
120
121
122
123
124
125
126
127
128 = 8^2 + 8^2
129
130:
137 = 4^2 + 11^2
138
139
140 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 (sum of the first seven squares)
140 = 2^2 + 6^2 + 10^2
150:
160:
168 = 2^2 + 8^2 + 10^2
169
170:
180:
181 = 9^2 + 10^2 ......................................................... 181^2 = 19^2 + 180^2
190:
191
192
193
194 = 5^2 + 13^2
194 = 1^2 + 7^2 + 12^2 = 3^2 + 4^2 + 13^2 = 3^2 + 8^2 + 11^2 = 5^2 + 5^2 + 12^2 = 7^2 + 8^2 + 9^2
200
201
202 = 9^2 + 11^2
216 (=6^3) = 3^3 + 4^3 + 5^3
242 = 11^2 + 11^2
243
244
245 = 7^2 + 14^2
258 = 59 + 61 + 67 + 71 (sum of four consecutive prime numbers)
269 = 10^2 + 13^2 ............... 269^2 = 69^2 + 260^2
270
271 = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43
300:
320 = 8^2 + 16^2
330
340
341
342
343 = 7^3 = 18^0 + 18^1 + 18^2
344
345
369 = 12^2 + 15^2
352 = 8^2 + 12^2 + 12^2
385 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2
390
391
392
393
394
395
400
450 = 3^2 + 21^2 ............ = 223 + 227 (sum of two consecutive prime numbers)
451
452
453
454
455
456 = 107 + 109 + 113 + 127 (sum of four consecutive prime numbers)
457 = 149 + 151 + 157 (sum of three consecutive prime numbers)
458
459
460
461 = 444 + 6 + 11
462 = 67 + 71 + 73 + 79 + 83 + 89 (sum of six consecutive prime numbers)
463 = 53 + 59 + 61 + 67 + 71 + 73 + 79 (sum of seven consecutive prime numbers)
464
465
466
467
468 = 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67
(sum of ten consecutive prime numbers)
469
470
480
481 = 15^2 + 16^2 (sum of 2 consecutive integers)
482
483
484
485
486 = 3^5 + 3^5
487
488
489 is an octahedral number.
490 = 7^2 + 21^2 ........... = 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5
491
492
493
494
495
496 is the third perfect number
496 = 1^3 + 3^3 + 5^3 + 7^3
496 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 (A triangular number)
497 = 89 + 97 + 101 + 103 + 107 (sum of consecutive primes)
498
499 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 (sum of consecutive primes)
500
501
502
503
504
505
506 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2
520 = = 6^2 + 22^2 = 14^2 + 18^2 .................... = 257 + 263 (sum of two successive prime numbers)
521
522 = 9^2 + 21^2 ............... = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5
523
524
525
548 = 8^2 + 22^2
549
550
551
552
553 = 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79
554
555
556
567
568
569
560 = 4^2 + 12^2 + 20^2
570
580 = 2^2 + 24^2 = 16^2 + 18^2
581 = 191 + 193 + 197
600:
601
602
603
605
606
607
608
609
610
611
612
613
614
615 = 9^2 + 10^2 + 11^2 + 12^2 + 13^2 (sum of five consecutive squares)
616
617
618
619
620
621
622
623
624
625
626 = 1^2 + 25^2 .... 626 = 5^4 + 1^4
627
628
629
630
640
650
660
661
662
663
664
665
666 = 3^6 - 2^6 + 1^6
666 = 6^3 + 6^3 + 6^3 + 6 + 6 + 6
666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2
670
671
672 = 4^2 + 16^2 + 20^2
673
674
675
700:
713 = 233 + 239 + 241
725 = 7^2 + 26^2 = 10^2 + 25^2 = 14^2 + 23^2 ... 725^2 = 333^2 + 644^2 = 364^2 + 627^2
736 = 4^2 + 12^2 + 24^2
737
738
739
740
750:
776 = 10^2 + 26^2
799 = 3^5 + 3^5 + 3^5 + 2^5 + 2^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5
800:
801 = 15^2 + 24^2
801 = (7! + 8! + 9! + 10!)/(7 * 8 * 9 * 10)
810
811 = 151 + 157 + 163 + 167 + 173
(sum of five consecutive primes)
811 (prime of the form 4n + 3) = 405 + 406
812
813
814
815
820 = 6^2 + 28^2 = 12^2 + 26^2
829 = 10^2 + 27^2 .............. 829^2 = 540^2 + 629^2
842 = 1^2 + 29^2
850:
859 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 3^5
860
861
862
863
864
865
866 = 5^2 + 29^2
881 = 16^2 + 25^2 .................................... 881^2 = 369^2 + 800^2
900
901 = 1^2 + 30^2 = 15^2 + 26^2
902
903
904
905
906
907 = 3^2 + 13^2 + 27^2
917 = 173 + 179 + 181 + 191 + 193
918
919
920
921
922
923
924
925
926
927
928
929 = 20^2 + 23^2 .... 929^2 = 129^2 + 920^2 .... 929^3 = 69^3 + 447^3 + 893^3
930
945 = 472 + 473
945 = 314 + 315 + 316
945 = 187 + 188 + 189 + 190 + 191
945 = 155+156+157+158+159+160
945 = 132+133+134+135+136+137+138
945 = 101+102+103+104+105+106+107+108+109
945 = 90+91+92+93+94+95+96+97+98+99
945 = 61+62+63+64+65+66+67+68+69+70+71+72+73+74
945 = 56+57+58+59+60+61+62+63+64+65+66+67+68+69+70
945 = 44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60+61
945 = 35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55
945 =22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48
945 = 17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46
945 = 10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44
945 = 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43
949 = 7^2 + 30^2 = 18^2 + 25^2
966 = 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139
(sum of eight consecutive primes)
980
981
982
983
984 = 8 + 88 + 888
985
986
987
988
989
990
991
992 = 8 + 8 + 88 + 888
993
994
995
996
997
998
999
1000 = 8 + 8 + 8 + 88 + 888
1115 = 103 + 107 + 109 + 113 + 127 + 131 + 139 + 149
1201 = 24^2 + 25^2 ............................. 1201^2 = 49^2 + 1200^2
1233 = 12^2 + 33^2
1233 is the smallest 2n-digit number that is the sum of the squares of its n-digit halves
1301 = 25^2 + 26^2 ............................. 1301^2 = 51^2 + 1300^2
1412 = 16^2 + 34^2
1717 = 6^2 + 41^2 = 14^2 + 39^2 ..... 1717^2 = 492^2 + 1645^2 = 1092^2 + 1325^2
1900 = 223 + 227 + 229 + 233 + 239 + 241 + 251 + 257
1927 = 2^11 – 11^2
1949 = 10^2 + 43^2 ............................................ 1949^2 = 860^2 + 1749^2
2000 = 8^2 + 44^2 = 20^2 + 40^2
2001
2002
2003
2004
2005
2006
2007
2008
2009 = 28^2 + 35^2
2010
2100
2200
2300
2357 = 26^2 + 41^2 ................................................ 2357^2 = 1005^2 + 2132^2
2357 = 773 + 787 + 797 ............................. 2357 = 461 + 463 + 467 + 479 + 487
2400
2401 = 7^4
2500
2501 = 1^2 + 50^2 = 10^2 + 49^2 ......... 2501^2 = 100^2 + 2499^2 = 980^2 + 2301^2
3000
3025 = 55^2 = 44^2 + 33^2
4000
4253 = 38^2 + 53^2 ............................................ 4253^2 = 1365^2 + 4028^2
5000
5008 = 48^2 + 52^2
5557 = 9^2 + 74^2 ..... 5557^2 = 1332^2 + 5395^2
2 + 3 + 5 + 7 + 11 + . . . + 3833 = 3847 + 3851 + . . . + 5557
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