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• Jan 16, 2009
Spoiler warning.

> 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
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