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Old 2008-06-26, 06:48   #1
devarajkandadai's Avatar
May 2004

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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

[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.
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