20080626, 06:48  #1 
May 2004
100111100_{2} Posts 
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. 
20080626, 14:26  #2 
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,889 Posts 

20080626, 21:36  #3 
∂^{2}ω=0
Sep 2002
República de California
2^{2}·2,939 Posts 
I simply saw the dreaded phrase "failure function" and didn't need to read further to know this was yet another "Look at me! I came up with another completely useless and utterly uninteresting integer sequence which I bothered to inflict on Neal Sloane!" thread.
Saying "this stuff is about as interesting as watching paint dry" would be an insult ... to drying paint. 
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