RSP Palindromes

     I like to play chess on the internet. It is often the case that players are rated with numbers according to how well they perform. Recently I noticed an interesting bit of number trivia about my rating in a certain type of chess. It said that I had 1661 points! (Not bad, but not the best of the players.)

     Of course, I was happy, because it was a palindrome. But upon looking more closely, it can be observed that 16 is a square number, and its reverse, 61. is a prime number! Moreover, this is unique for all squares from 1 to 100.

     So what do we have here? Well, WTM wants to call something like this a Reversible-Square-to-Prime Palindrome, or RSP Palindrome, for short.

     Here is a chart of all numbers less than 100 (with one exception) that produce RSP Pals.

n	Square	Prime	Palindrome	Prime Factors
4	16	61	1661	11 x 151
14	196	691	196691	11 x 17881
19	361	163	361163	11 x 32833
28	784	487	784487	11 x 71317
32	1024	4201	10244201	11 x 127 x 7333
37	1369	9631	13699631	11 x 1245421
38	1444	4441	14444441	11 x 17 x 77243
41	1681	1861	16811861	113 x 17 x 743
62	3844	4483	3844483	7 x 112 x 45389
85	7225	5227	72255227	11 x 6568657
89	7921	1297	79211297	11 x 127 x 56701
95	9025	5209	90255209	11 x 79 x 283 x 367
97	9409	9049	94099049	11 x 232 x 103 x 157

     Now, dear reader, you are invited to continue this list. Send any results you find and WTM will post them here.

     As to the exception referred to above... 40² = 1600. The reverse of 1600 is either 0061, or 61 if the leading zeros are suppressed. This gives us 16000061 and 160061 as two more RSP Pals for this range.

     Next, the curious thinker should be asking himself... what about cubes and their reversals? Do similar cases exist for RCP Palindromes? The answer is not long in coming to light. Observe:

5³ = 125      521 is prime.      Hence 125521 is a RCP Pal.

     Except for the trivial 50³ case, how many RCP Palindromes can be found for n < 100?