Max <[EMAIL PROTECTED]> wrote: On 2/17/06, Farideh Firoozbakht wrote:
> > Can anybody prove that P(p) can never be prime for prime p or
> > find prime P(p) for some prime p (in which case p must be > 150000) ?
>
> Note that P(2)=7 & P(3)=43, so I think your question should be:
>
> Can anybody prove that P(p) can never be prime for prime p, p>3 or
> find prime P(p) for some prime p>3 (in which case p must be > 150000) ?
Thanks for pointing this out!
Yes, p must be greater than 3. I forgot to mention that.
btw, the problem has an algebraic origin as it appeared in research
related to PU_3(q) groups. And the author of original message about
this problem is very much interested in getting the answer.
Max
Max,
We can easily prove the following nine assertions.
1. If Mod[n,3]=2 then 7 divides P(n).
2. If Mod [n, 60] is a member of the set {7,9,19,21,31,33,43,45,55,57} then
13 divides P(n).
3. If Mod [n,90] is a member of the set {5,8,23,26,41,44,59,62,77,80} then
19 divides P(n).
4. If Mod [n,180] is a member of the set {28,34,64,70,100,106,136,142,172,178}
then 37 divides P(n).
5. If Mod [n, 180] is a member of the set {28,38,88,98,148,158} then 61
divides P(n).
6. If Mod [n, 330] is a member of the set
{47,58,113,124,179,190,245,256,311,322}
then 67 divides P(n).
7. If Mod [n, 390] is a member of the set
{31,70,109,148,187,226,265,304,343,382}
then 79 divides P(n).
8. If Mod [n, 510] is a member of the set
{47,98,149,200,251,302,353,404,455,506}
then 103 divides P(n).
9. If Mod[n,14]=3 then 43 divides P(n).
If p is a prime greater than 3 and P(p) is prime then by using the assertions
1& 2
Mod[p, 60] instead one the twelve numbers
1,7,11,13,17,19,23,29,31,37,41,43,47,
49,53 & 59 is one of the four numbers 1,13,37& 49. Also by using all the
assertions
1,2,3,5,6,7,8 & 9 Mod[p,lcm(3,14,60,90,330,390,510)]=Mod[p, 3063060] is
restricted
to a set of 105600 numbers instead of the set A={k|
gcd(k,3063060)=1,k<3063060} ,
note that length(A) =552960.
So for a prime p the probability that P(p) is prime isn't greater than105600/
552960=
55/288=0.190972
Farideh
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