Satiric Hue: Achier Suit? ==> Rice Hiatus!
A Heuristic
Notation: p represents any prime in the sequence A 002496.
p' represents a prime number in the seq A141293
References: [1] Definition of failure function(Polynomial) (Maths EncyclopediaPlanetmat.org).
[2] Examle failure function (polynomial) "

This pertains to the conjecture that there are infinitely many primes of the form x^2 +1.
Let P be the largest known prime number of the form x^2 +1. Let x_0 be the value of x such
that x_0^2 + 1 = P.
1) Now let phi(x) = x^2 + 1. phi(x + kP) is congruent to 0 (mod P) since x = psi(x_0)
= x_0 + kP .. ...vide [1] above.
(Here k belongs to Z).
Now consider the discrete interval x_0 to x_0 + P. All composite numbers of the form
x^2 + 1, where x is an integer in the interval x_0 to x_0 + P, must have necessarily satisfied
one of the failure functions 1 + 2k, 2+ 5k, 3 +10k, 4 + 17k......This is because whenever
x^2 + 1 is composite one of its factors is less than x. This also implies that all failures
in this interval have the basic structure 2^kp_1^kp_2^k.p'^kp'^2p'^3..... (k belongs to W and is unbounded
excepting in the case of 2 where k can assume only the value 0 or 1).
It must be understood that when there are one or more values of x in this interval not
satisfying any of the prior failure functions, including the second order failure functions
( ref [2] above) they are such that phi(x) is prime, that P represents the largest of these,
the relevant value of x is represented by x_0 and the the new longer discrete interval is the new x_0
to x_o + P.
We must bear in mind the fact that the members of seq A 140 687, including Mersenne primes,
do not contribute a single failure function thus increasing the probabality of leaving
values of x in the new interval uncovered by the failure functions and thus increasing the
probabality of there being infinitely many primes of the form x^2 + 1.
If at all the lengthening interval x_0 to x_0 + P is discovered to be completely covered
by the prior failure functionsit can only mean the following:
The discrete intervals x_0 to x_0 + P,x_0+ P to x_o+2P, x_0 +2P to X_0 + 3P.......
all have members exhibiting a basic identical structure 2^kp_1^kp_2^k....P^kfollowed by a string of p', not relevant to the proof.
It is only the string of ps that is recurrent ( of course k, the variable exponent may increase)
There seems to be only two possibilities:
a) The interval x_0 to x_0 + P is completely covered by the primary and secondary failure functions
(thoses generated by p and those generated by p' resply.). Since x^2 +1 is a strictly monotonic increasing function of x
all the composite numbers generated after x0 + P have a string of ps of a certain maximumum length
(and P or a power of P appearing periodically followed by a growing string of p's).
b) The interval x_0 to x_0 +P is ever growing.
My gut feeling is that b above is true. In fact I will not be surprised that any irreducible
quadratic expression in x will generate an infinite set of prime numbers having the shape of the quadratic.
Perhaps programming can settle the issue.
