Problems & Puzzles: Puzzles
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Problems & Puzzles: Puzzles
 From March 21-28, 2026, contributions came from Michael Branicky, Carlos Rivera, Gennady Gusev, Emmanuel Vantieghem *** Michael Branicky wrote: Q2:Primes starting at position 1 in the digits of Pi is OEIS A005042; lengths are in A060421.Thus, the largest known (probable?) prime starting at position 1 has 613373 digits (from Adrian Bondrescu, May 29 2016) Q3:Using OEIS A047658, the largest known prime starting at position 2 has 16350 digits. I found a prime (PRP) with 31001 digits starting at position 34534 in Pi: 794040...47272 Q4 & Q5 (for Q2): Testing all the primes from Q2:- the largest known emirp starting at position 1 is 314159;- the largest known palprime starting at position 1 is 3.... For Q5/Q3, the largest palprime I found was 176860696068671 (15 digits) starting at position 298503034.This was already known in Prime Curios: https://t5k.org/curios/page.php?number_id=22424 But https://academic.oup.com/imrn/article/2024/18/12466/7729537 (also at https://arxiv.org/pdf/2307.16637 ) cites two sources forthe palprime 9609457639843489367549069 in Pi. I could not find theindex listed. But my own search shows it does not appear in thefirst 10^9 terms. For Q4/Q5, it is easy to find large emirps within the first 10^9 digits of pi,e.g., there is one with 1100 digits starting at position 37611: 9328040171...5323650893 (1100 digits), . Found by M. Branicky For Q4/Q3, there is a 5000-digit emirp starting at position 3244207: 109159...322501. Found by M. Branicky. *** Carlos Rivera wrote: A.- I found two more public pages, other than the six listed above, to search integer sequences inside pi: g) https://calculat.io/en/number/search-sequence-in-pi h) https://www.dcode.fr/pi-digits B.- This is what I found: a 19 digits palprime with index, author, public-page and date (Aug 8, 2024(?)). https://github.com/fontes-mrc/pi-palindromes Palindrome: 912-501-055-0-550-105-219 (9,125,010,550,550,105,219) 19 digits. "nine quintillion, one hundred twenty-five quadrillion, ten trillion, five hundred fifty billion, five hundred fifty million, one hundred five thousand, two hundred nineteen (US)*"Index:78,833,628,391, "seventy-eight billion, eight hundred thirty-three million, six hundred twenty-eight thousand, three hundred ninety-one (US)"Author: Mauricio Fontes * Numbers spelled in Enlish US, by https://www.ego4u.com/en/cram-up/vocabulary/numbers/generator BTW, I (CR) have not checked already totally this claim. As a matter of fact 9,125,010,550,550,105,219 is Palprime but I have not checked the reported index inside pi: None of the eight reported tools by me for seeking an integer inside pi has ben successful with this 19 digits palprime. Q6. Regarding the palprime 9609457639843489367549069 (25 digits) inside pi, mentioned in several places in the web, for example, I was unable to find serious author(s) neither the index position inside pi. In short, CharGPT wrote: "No authoritative source credits a specific person with discovering that exact palindromic prime inside π, nor is there a well-documented “first discovery” date." *** Gennady wrote: My modest results: Q2. Prime: from 1st digit to 16350. Q3. Prime: from 20th digit to 17174. Q4. Emirp: from 2976 to 5977. Q5. Palprime: 488-492 -> 38183. *** Emmanuel Vantieghem wrote: Q2. The biggest prime I found has 6205 digits. Confirmed prime by Alperton. Buzy to be proved prime by PRIMO (already more than 124 hours working). If there is a bigger prime, it will have more than 10000 digits (and will require a lot of time to be proved). Q3. I found a prime of 8573 digits, starting at position 8626. Confirmed prime by Alpertron. But I'll not ask PRIMO to prove the primality.I'm sure there exist much bigger ones. Q4. I found an emirp of 1008 digits, starting at position 2517776. But I'm sure there exist much bigger ones. Q5. My biggest palprime is 176860696068671 starting at position 298503033.No doubt there exist bigger ones. *** On April 1. 2026 Ashaz Jameel wrote:
For q5,3) there are 2 palprimes in the first 100 trillion digits of pi
of size 25
9609457639843489367549069 (25 digits) at pos
33135773126758
7331530558321238550351337
(25 digits)
at pos 33044988112960
taken from https://github.com/tvbgo/pi-
The author of the pi-palprimes is Tomas Gomes, from Rio de Janeiro, who got these results around Set. 25, 2022, according to the link provided by Jameel. This message from Jameel also answers the question posed at the beginning of this puzzle, nmely Q6: Who is the author of the pipalprime 9609457639843489367549069?... Hopefully this claim is absolutely true. ... Primes verified by CR using ECM by Darío Alpern. Positions has not been verified by me. ***
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