Spoiler warning.

David Broadhurst wrote:

> p1 is the smallest titanic prime of the form

> sqrt(3*x^2-2) with integer x.

>

> p2 is the smallest titanic prime of the form

> sqrt(5*y^2-4) with integer y.

>

> p3 is the smallest prime such that

> 2*p1*p2*p3-1 is prime.

>

> Find a metrical connection between p3 and liberty.

I suck at diophantine equations so I didn't even try but just

bruteforced the first few x values which gave integer square root.

A quick lookup of the x sequences in OEIS and there they were.

http://www.research.att.com/~njas/sequences/A001835 has comment:

Terms are the solutions to: 3x^2-2 is a square.

http://www.research.att.com/~njas/sequences/A001519 has comment:

Terms for n>1 are the solutions to : 5x^2-4 is a square.

A little PARI+PFGW later and I get a 1673-digit p1,

a 1223-digit p2, and p3 = 12241.

The metrical connection is left as an exercise for the reader

(actually, I don't know and didn't look hard).

I haven't run Primo on p1 and p2. Knowing David there might be

easy proofs but I'm not looking. OK, I'm lazy.

Assuming they are prime, PFGW proved 2*p1*p2*p3-1.

--

Jens Kruse Andersen