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 & nonpalindrome
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
200 42398 (9)_{195}
57601 10675(9)_{195}
300 269174 (9)_{294}
730825 528037(9)_{294}
400 14462 (9)_{395}
85537 73558(9)_{395}
500 36530 (9)_{495}
63469 96436(9)_{495}
608 34652 (9)_{603}
65347 74356(9)_{603}
704 22568 (9)_{699}
77431 13477(9)_{699}
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)_{850}2047101(0)_{850}1 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 1997record. This is the one I got after one week of
search with a little code in Ubasic: 1(0)_{1992}7084987
& 7894807(0)_{1992}1
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 N1 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^n1 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.
10^39291+(428375201999999999)*10^1960 is a
reversible prime with 3929 digits. 3929 is also a reversible prime.
10^20031+(k999999999)*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.
It has the form 1 0(4998) 941992101 0(4998) 1. The reverse prime is
10^10006+101299149*10^4999+1. PrimeForm/GW found it after only 2.2% of
the expected prp tests. It has 10007 digits and 10007 is a reversible
prime.
*** On March 2011, J. K. ndersen wrote:
p = 10^5013+10^3296+10^1834+1 is a reversible prime.
reverse(p) = 10^5013+10^3179+10^1717+1. PrimeForm/GW found and proved the
primes.
A reversible prime is also called an emirp (prime backwards).
Similarly, a reversible semiprime is called an emirpimes (semiprime
backwards).
The largest known palindromic prime is q = 10^200000+47960506974*10^99995+1,
found by Bernardo Boncompagni:http://primes.utm.edu/primes/page.php?id=94993.
p*q with 205014 digits is the largest known emirpimes.
p was constructed such that reverse(p*q) = reverse(p)*q.
***
