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{{DISPLAYTITLE:Palindromic Prime Triplet}}<br />
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''This conversation was exported from ChatGPT''<ref>{{cite web|title=Palindromic Prime Triplet|url=https://chatgpt.com/share/694fa129-5c4c-800e-b887-30cd25632750|publisher=ChatGPT|access-date=2025-12-28}}</ref><br />
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== gpt-5-t-mini ==<br />
=== User: Bob is thinking of 3 distinct primes. ===<br />
Bob is thinking of 3 distinct primes. Their sum is less than 30 and their concatenation is a palindrome. He asks Jane what his primes are. Jane suggests: 3,11,13-- a triplet whose sum is 27. Bob rejects her answer. Bob's response is justified. Without using arithmetic, what's the most likely explanation for this situation?<br />
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=== Assistant: Write the three primes as digit-strings A,B,CA,B,CA,B,C and look at the string ABCABCABC. ===<br />
Write the three primes as digit-strings A,B,CA,B,CA,B,C and look at the string ABCABCABC. If that string is a palindrome, reading it backwards gives<br />
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reverse(C) reverse(B) reverse(A)=A B C.\text{reverse}(C)\,\text{reverse}(B)\,\text{reverse}(A)=A\,B\,C.reverse(C)reverse(B)reverse(A)=ABC.<br />
For that equality to hold the middle block must read the same forwards and backwards, so BBB itself is a palindromic prime. (Otherwise the reversed middle block would put different digits into the centre of the whole string.)<br />
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So BBB must be a palindromic prime (the only two-digit palindromic prime is 11; single-digit primes are palindromes too). The outer blocks must be digit-reverses of one another (their digits mirror around the centre). That means you cannot have one outer prime be single-digit and the other two-digit (a two-digit number’s reverse is two-digit), so the obvious choice for distinct outer primes is a pair like 13 and 31.<br />
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Jane’s proposal 3,11,133,11,133,11,13 violates that forced digit-structure: a single digit 3 cannot be the digit-reverse of the two-digit 13, so it cannot be the correct triple arising from the palindrome condition Bob stated. Hence Bob is justified in rejecting her guess.<br />
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== References ==<br />
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[[Category:ChatGPT Shares]]<br />
[[Category:openai]]</div>Lukegao