CONTEST++ two more conjectures, more $$
- I offer a $51 prize to the first person who can submit
a verifiable counterexample by New Year's day
either for the following conjectures.
(x,A,B,c,k,f : integers)
Let A = 20x^2 + 10x + 1;
Let B = 10x^2 + 4x + 1;
Let c = trunc(A/sqrt(5)) - 1;
1) For any x > 0, apply the issquare test to each k
in the interval c <= k < B.
If there exists a value of k that satisfies the
conditions of the test, then A is composite,
and there will be at least one value of k such
that 2*k - 1 will have a factor in A.
(issquare test: 0 < 5*(2*k -1)^2 - 4*A^2 = f^2)
2)For any x > 0, then if A and B are placed in the formula
below, the five values can be used to make a polynomial
or a set of difference equations that will generate a
fourth degree sequence.
A , A + 1*B*10 , A + 4*B*10 + 60,
A + 9*B*10 + 360, S + 16* B*10 + 1200
The result will always be either a pure +1 mod 10 sequence
(all prime factors end in one) or a defective +1 mod 10
sequence ( all prime factors end in one except those
containing a finite number of squares of primes that
end in nine):
As I really want these questions settled, I would be happy
to assist any interested person.