"Rare" Primes

[Mini-Geek]

Originally Posted by R.D. Silverman

Although not explicitly stated, I believe that the domain is N. Now,
f(x) is prime for x = 0, 4 and no other. If you accept the more general
definition of prime (i.e. not restricted to just N) then f(x) will be prime
i.o. (although a proof is lacking). If we allow x \in R, then f(x) is
indeed prime the required number of times.

If we allow real numbers as x, then f(x) is prime 19 times - 9 times for negative x values as f(x) increases, once for x=0 where f(x) levels off at 29, and 9 more times for positive x values as f(x) decreases. There would be 10 unique primes, but 19 values of x that produce a prime f(x).

[retina]

Originally Posted by R.D. Silverman

Although not explicitly stated, I believe that the domain is N. Now,
f(x) is prime for x = 0, 4 and no other. If you accept the more general
definition of prime (i.e. not restricted to just N) then f(x) will be prime
i.o. (although a proof is lacking). If we allow x \in R, then f(x) is
indeed prime the required number of times.

Thanks, yeah I was assuming x is an element of R.

f(x)=398-x^2; x element of N

To satisfy Mini-Geek's requirement of total primes: f(x)=143-x^2; x element of Z

[Housemouse]

I am obviously not communicating well. However you could have referred me to primes.utm.edu (as done by Cruelty) or Wikipedia or some other helpful web site. I am looking for "interesting" examples as listed on these web sites.

On the primes.utm.edu site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n-2+1.

Has it been proven that only 7 exist?

Wikipedia, states that there is only one positive Genocchi prime; has this been proven?

[retina]

Originally Posted by Housemouse

However you could have referred me to primes.utm.edu (as done by Cruelty) or Wikipedia or some other helpful web site.

It is usually assumed that you know how to google things you are interested in.

[CRGreathouse]

Originally Posted by Housemouse

On the primes.utm.edu site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n-2+1.

Has it been proven that only 7 exist?

No. Given that (10^n + 666) * 10^(n-2) + 1 is not divisible by 2, 3, or 5, a quick guess at the 'chance' it's prime as
15/4 * (1/log(10^(2n-2)))

The sum of this from 2 to 3000 is 6.98, so having 7 from n = 2 to 3000 is pretty much what you'd expect. The expected number up to a million is 11.72, so it would be unusual if only 7 existed. In fact, since the harmonic series diverges, you'd naively expect an infinite number of such primes.

Originally Posted by Housemouse

Wikipedia, states that there is only one positive Genocchi prime; has this been proven?

MathWorld has "D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.".

[CRGreathouse]

To the original question: Of course there are uncountably many sets of primes (beth_1), a countable number of which sets are finite; but the question seems to be about intuitively 'interesting' sets of primes.

Toward that end I suggest my small compilation here:
http://en.wikipedia.org/wiki/User:CR...special_primes

[Housemouse]

I found additional "interesting examples" at primes.utm.edu, under Rudolf Ondrejka's top ten.

Sometimes finding useful information using google is like looking for a needle in a hay stack.