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 
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)₈₅₀2047101(0)₈₅₀1

This prime has 1709 digits. Then it seems not very hard to reach and supersede this record.

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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?

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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.
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.

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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.

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