--- In

primenumbers@yahoogroups.com, "Jens Kruse Andersen"

<jens.k.a@...> wrote:

> I now get 1673-digit p1, 1001-digit p2, and p3=151.

Congratulations on completing the maths.

> I still don't know what the metric connection is.

Inaugural puzzle:

> 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.

Solution:

1) p1 = 3*lucasU(4,1,2925) - lucasV(4,1,2925)/2

is the smallest titanic prime in the integer sequence

http://www.research.att.com/~njas/sequences/A001834
with a[1] = 1, a[2] = 5, a[n] = 4*a[n-1] - a[n-2],

ensuring that a[n]^2 = 3*x^2 - 2, with integer x.

Then a[n] = 3*lucasU(4,1,n) - lucasV(4,1,n)/2 solves

the recursion, in terms of generalized Lucas numbers.

PFGW test:

Calling N-1 BLS with factored part 4.90%

and helper 1.80% (16.51% proof)

3*lucasU(4,1,2925)-lucasV(4,1,2925)/2

is Fermat and Lucas PRP!

Primo timing:

Total=2h 11mn 27s

DecimalSize=1673

2) p2 = (5*F(2*2394) - L(2*2394))/2

is the smallest titanic prime in the integer sequence

http://www.research.att.com/~njas/sequences/A002878
with a[1] = 1, a[2] = 4, a[n] = 3*a[n-1] - a[n-2],

ensuring that a[n]^2 = 5*y^2 - 4, with integer y.

Then a[n] = (5*F(2*n) - L(2*n))/2 solves the recursion,

in terms of the Fibonacci and classical Lucas numbers.

PFGW test:

Calling N+1 BLS with factored part 7.61%

and helper 3.79% (26.69% proof)

(5*F(2*2394)-L(2*2394))/2

is Fermat and Lucas PRP!

Primo timing:

Total=19mn 41s

DecimalSize=1001

3) p3 = 151

is the smallest prime such that 2*p1*p2*p3-1 is prime.

PFGW test with helpers p1 and p2:

Calling N+1 BLS with factored part 100.00%

and helper 0.65% (300.68% proof)

151*(3*lucasU(4,1,2925)-lucasV(4,1,2925)/2)*(5*F(2*2394)-L(2*2394))-1

is prime!

4) Then by googling

> 151 prime

we are directed, at very first choice, to Chris Caldwell's page

http://primes.utm.edu/curios/page.php/151.html
reminding us of the Statue of Liberty.

I reflected on the poem inscribed within its pedestal

http://www.libertystatepark.com/emma.htm
which served to remind me of a time when the

United States of America were a better nation

than that belittled so wantonly on 20/03/03

by the ignominious Bush and his shameless toady, Blair:

http://tech.groups.yahoo.com/group/primenumbers/message/11863
http://tech.groups.yahoo.com/group/primenumbers/message/11875
http://tech.groups.yahoo.com/group/primenumbers/message/11883
http://tech.groups.yahoo.com/group/primenumbers/message/12042
Let us dare to hope for some improvement:

> There is a call to life a little sterner,

> And braver for the earner, learner, yearner.

David