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, see but not the best of the players.)
Of course, try 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.
|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:
Except for the trivial 503 case, how many RCP Palindromes can be found for n < 100?