Problems & Puzzles: Puzzles Puzzle 20.- Reversible Primes 13 is the least reversible non palindrome prime because 31 is also a prime. I’ll keep here a list of the 10 largest reversible & non-palindrome primes. As a maybe good starting point I have produced a few reversible primes from 200 to 704 digits. N=10^E -1 -Z E
Z
N
Nrev Can you produce higher reversible primes (of course they need not to be of this type)?
Reading one more time the Rudolph
Ondrejka Primes Collection I saw recently that there is a record for
this kind of primes established by H. Dubner in 1997: 1(0)8502047101(0)8501 This
prime has 1709 digits. Then it seems not very hard to reach and supersede
this record. ***
Just while other people get interested in
these prime I spent a week getting a larger reversible prime than the
current Dubner's 1997-record. This is the one I got after one week of
search with a little code in Ubasic: 1(0)19927084987
& 7894807(0)19921
are reversible primes, 2000 digits each (29/6/2001). These
numbers are expressed: 10^1999+7084987 & 7894807*10^1993+1,
respectively. Of course that only the
second is rigorously a prime (according to the N-1 test of PRIMEFORM). The
first one is up to day a strong probable prime. Does
anybody wants to get the rigorous primality certificate of the first one,
using TITANIX? ***
Great news! They came from J. K.
Andersen (June 2003). His results and his method are very interesting: I have found the 10 largest known reversible primes. If a search starts with 10^n+1 or 10^n-1 and only
changes the middle digits then the solutions with probable primes become
easily provable. This is important since titanic prp solutions can be
found faster than Primo (formerly Titanix) could prove them. 10^3929-1+(428375201-999999999)*10^1960 is a
reversible prime with 3929 digits. 3929 is also a reversible prime. 10^2003-1+(k-999999999)*10^997 is a reversible prime
with 2003 digits for the following nine k: 107671501, 158957701,
168586801, 268720301, 292300601, 318811301, 689715601, 856978001,
996358669. In each case the decimal prime and the reverse is k
and reverse(k) with 997 9's on both sides. I trial factored with my own C program using Michael
Scott's Miracl library. prp tests were made with PrimeForm/GW which also
proved all the primes. Later on October 13, 07 he added: 10^10006+941992101*10^4999+1 is a proven
gigantic reversible prime. Note
by CR: I checked both primes with
ECM
by Dario Alpern and , true, these two integers are primes. *** On March 2011, J. K. Andersen wrote: p = 10^5013+10^3296+10^1834+1 is a reversible prime. *** According to a Wikipedia
page, and the following
curio, since this last Feb. 16, 2026, "The largest known
emirp as of 2026 is 3,867,632,931 × 10^10001 + 1 discovered by Stephan Schöler".
It has 10011 digits, beating the previous record by J. K. Andersen with
10007 digits, reported on October 13, 2007. See above.
Note
by CR: I checked both primes with
ECM
by Dario Alpern and, true, these two integers are primes.
***
On March 17, 2026, Simon Cavegn wrote:
Found two new record emirps with 15307 digits.
The number of digits is an emirp (15298+9= 15307)
The number of zeros is also an emirp.(15298-9 = 15289)
Both of them have been verified by me using
ECM
by Darío Alpern.
***
On March 18, 2026, Simon Cavegn wrote again:
Seems two more emirps were found by other people as well:
The first link is about a 10069 digits emirp
The second link is about a 12346 digits emirp
Both of them have been verified by me using
ECM
by Darío Alpern.
***
On Mach 19, Simon Cavegn wrote again:
Dear Carlos Rivera,
10^20000 + 518406362 * 10^9996 + 1
Original from:
http://factordb.com/index.php?id=1100000008642430228
(I checked it.)
Simon Cavegn
(CR,
I made my own verification using
ECM
by Darío Alpern.)
***
* Both of them have been verified by me (CR)
using
ECM
by Darío Alpern.
***
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