20080626, 06:48  #1 
May 2004
2^{2}·79 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 
Nov 2003
7460_{10} Posts 

20080626, 21:36  #3 
∂^{2}ω=0
Sep 2002
República de California
2^{3}·1,459 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. 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Rice Elevator Video Sent to NFL 3 Months Ago  kladner  Soap Box  15  20151107 16:22 
Back from a short hiatus...  mdettweiler  No Prime Left Behind  2  20090220 16:26 
Hiatus  ValerieVonck  Octoproth Search  0  20080119 10:40 
Hiatus  delta_t  Marin's Mersennearies  1  20051222 13:03 