## Palindrome 2662

We may partition the palindrome 2662 into two integers, one of which is divisible by 29 and the other by 43.

$2662 \; = \; (29\times 31) \; + \; (41\times 43)$

29, 31, 41, 43   are prime numbers

29 and 31 form a twin prime pair.
41 and 43 form a twin prime pair.
29 is a Sophie Germain prime
41 is a Sophie Germain prime

2662   is of the form   $p(p + 2) + q(q + 2)$

Find other palindromes that can expressed in this fashion.

David found:

$(17\times 19) \; + \; (149\times 151) \; = \; 22822$

17 and 19 form a twin prime pair.
149 and 151 form a twin prime pair.

$(269\times 271) \; + \; (1607\times 1609) \; = \; 2658562$

269 and 271 form a twin prime pair.
1607 and 1609 form a twin prime pair.

$(179\times 181) \; + \; (239\times 241) \; = \; 89998$

179 and 181 form a twin prime pair.
239 and 241 form a twin prime pair.
179 is a Sophie Germain prime
239 is a Sophie Germain prime

$(2549\times 2551) \; + \; (4721\times 4723) \; = \; 28799782$

2549 and 2551 form a twin prime pair.
4721 and 4723 form a twin prime pair.
2549 is a Sophie Germain prime

Who was the last significant mathematician to define one as a prime?
https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf

math grad - Interest: Number theory
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### 5 Responses to Palindrome 2662

1. David @InfinitelyManic says:

Some…
(43 x 53) + (73 x 83) = (2279 + 6059) = 8338
(43 x 53) + (67 x 97) = (2279 + 6499) = 8778
(41 x 61) + (73 x 89) = (2501 + 6497) = 8998
(41 x 47) + (59 x 71) = (1927 + 4189) = 6116
(37 x 41) + (61 x 79) = (1517 + 4819) = 6336
(31 x 61) + (79 x 83) = (1891 + 6557) = 8448
(31 x 61) + (71 x 97) = (1891 + 6887) = 8778
(31 x 47) + (61 x 89) = (1457 + 5429) = 6886
(31 x 43) + (53 x 67) = (1333 + 3551) = 4884
(31 x 37) + (73 x 97) = (1147 + 7081) = 8228
(29 x 71) + (73 x 83) = (2059 + 6059) = 8118
(29 x 43) + (47 x 61) = (1247 + 2867) = 4114
(29 x 31) + (73 x 79) = (899 + 5767) = 6666
(29 x 31) + (41 x 43) = (899 + 1763) = 2662
(23 x 53) + (59 x 83) = (1219 + 4897) = 6116
(23 x 41) + (61 x 83) = (943 + 5063) = 6006
(23 x 41) + (59 x 97) = (943 + 5723) = 6666
(23 x 31) + (67 x 79) = (713 + 5293) = 6006
(23 x 31) + (43 x 97) = (713 + 4171) = 4884
(23 x 31) + (43 x 53) = (713 + 2279) = 2992
(23 x 29) + (47 x 71) = (667 + 3337) = 4004
(19 x 79) + (83 x 89) = (1501 + 7387) = 8888
(19 x 59) + (71 x 97) = (1121 + 6887) = 8008
(19 x 47) + (53 x 67) = (893 + 3551) = 4444
(19 x 37) + (67 x 89) = (703 + 5963) = 6666
(19 x 31) + (37 x 59) = (589 + 2183) = 2772
(19 x 29) + (31 x 61) = (551 + 1891) = 2442
(17 x 61) + (73 x 97) = (1037 + 7081) = 8118
(17 x 43) + (83 x 89) = (731 + 7387) = 8118
(17 x 43) + (47 x 79) = (731 + 3713) = 4444
(17 x 41) + (53 x 79) = (697 + 4187) = 4884
(17 x 29) + (43 x 97) = (493 + 4171) = 4664
(17 x 29) + (43 x 53) = (493 + 2279) = 2772
(17 x 23) + (47 x 53) = (391 + 2491) = 2882
(17 x 19) + (47 x 83) = (323 + 3901) = 4224
(17 x 19) + (31 x 79) = (323 + 2449) = 2772
(17 x 19) + (23 x 73) = (323 + 1679) = 2002
(13 x 37) + (47 x 89) = (481 + 4183) = 4664
(13 x 31) + (37 x 67) = (403 + 2479) = 2882
(13 x 19) + (61 x 67) = (247 + 4087) = 4334
(13 x 19) + (59 x 73) = (247 + 4307) = 4554
(13 x 19) + (41 x 97) = (247 + 3977) = 4224
(13 x 17) + (31 x 61) = (221 + 1891) = 2112
(13 x 17) + (23 x 29) = (221 + 667) = 888
(11 x 13) + (17 x 29) = (143 + 493) = 636
(7 x 71) + (73 x 83) = (497 + 6059) = 6556
(7 x 47) + (79 x 83) = (329 + 6557) = 6886

2. David @InfinitelyManic says:

Of the form p(p+2) + q(q+2):

(269 x 271) + (1607 x 1609) = (72899 + 2585663) = 2658562
(179 x 181) + (239 x 241) = (32399 + 57599) = 89998
(29 x 31) + (41 x 43) = (899 + 1763) = 2662
(17 x 19) + (149 x 151) = (323 + 22499) = 22822
(1 x 3) + (827 x 829) = (3 + 685583) = 685586

• David @InfinitelyManic says:

At least one Sophie Germain prime <= 15000
(10859 x 10861) + (12821 x 12823) = (117939599 + 164403683) = 282343282
(9629 x 9631) + (13721 x 13723) = (92736899 + 188293283) = 281030182
(6131 x 6133) + (13709 x 13711) = (37601423 + 187964099) = 225565522
(2549 x 2551) + (4721 x 4723) = (6502499 + 22297283) = 28799782
(179 x 181) + (239 x 241) = (32399 + 57599) = 89998
(29 x 31) + (41 x 43) = (899 + 1763) = 2662
(1 x 3) + (827 x 829) = (3 + 685583) = 685586

3. paul says:

Here are a few more

17 x 19 + 149 x 151 = 22822
29 x 31 + 41 x 43 = 2662
179 x 181 + 239 x 241 = 89998
269 x 271 + 1607 x 1609 = 2658562
881 x 883 + 4649 x 4651 = 22400422
1721 x 1723 + 169319 x 169321 = 28672227682
2081 x 2083 + 149249 x 149251 = 22279897222
2141 x 2143 + 163019 x 163021 = 26580108562
2549 x 2551 + 4721 x 4723 = 28799782
2999 x 3001 + 8999 x 9001 = 89999998
3461 x 3463 + 49409 x 49411 = 2453333542
6131 x 6133 + 13709 x 13711 = 225565522
6359 x 6361 + 50591 x 50593 = 2600000062
7949 x 7951 + 143651 x 143653 = 20699099602
8999 x 9001 + 50591 x 50593 = 2640550462
9629 x 9631 + 13721 x 13723 = 281030182
10859 x 10861 + 12821 x 12823 = 282343282
12917 x 12919 + 149969 x 149971 = 22657875622
13829 x 13831 + 93479 x 93481 = 8929779298
13829 x 13831 + 155537 x 155539 = 24383338342
14591 x 14593 + 162389 x 162391 = 26583438562
15137 x 15139 + 155849 x 155851 = 24518381542
16139 x 16141 + 93479 x 93481 = 8999009998
18311 x 18313 + 46439 x 46441 = 2492002942
20357 x 20359 + 46769 x 46771 = 2601881062
20771 x 20773 + 168899 x 168901 = 28958685982
24419 x 24421 + 154211 x 154213 = 24377677342
28277 x 28279 + 167339 x 167341 = 28802320882
33329 x 33331 + 154211 x 154213 = 24892229842
52901 x 52903 + 58367 x 58369 = 6205445026
80831 x 80833 + 127079 x 127081 = 22683138622
81047 x 81049 + 117809 x 117811 = 20447974402
86627 x 86629 + 145679 x 145681 = 28727072782
93251 x 93253 + 110879 x 110881 = 20990309902
106961 x 106963 + 115979 x 115981 = 24892229842
114689 x 114691 + 117041 x 117043 = 26852625862
135281 x 135283 + 223841 x 223843 = 68406460486
149111 x 149113 + 209567 x 209569 = 66153135166
154277 x 154279 + 211151 x 211153 = 68386868386
155081 x 155083 + 195047 x 195049 = 62094149026
162527 x 162529 + 185531 x 185533 = 60837473806
163307 x 163309 + 193871 x 193873 = 64255855246

Paul.