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[P] Application of failure functions- sketch proof. by akdevaraj Jul 5
The object of this message is to demonstrate the application of failure functions in sketching a possible proof of the conjecture that there are infinitely many primes of the form x^2 + 1. Let p_0 be the largest known prime of form x^2 + 1. Let x_0 be the value of x such that x_0^2 + 1 = p_0. Consider the interval x_0, x_0 + p_0. (This interval is chosen since (x_0 + p_0)^2 + 1 is congruent to 0 mod (p_0)). Iteration 1: Let x_1 be the largest value of x in the above interval not covered by the relevant failure functions viz x = 1 + 2*k, 2 + 5*k, 4 + 17*k, 6 + 37*k ...... x_1^2 + 1 is prime (see message: application of failure functions ). Let p_1 be this prime i.e. x_1^2 + 1 = p_1. Note that p_1 is greater than p_0.Iteration 2: consider the interval x_1, x_1 + p_1. Repeat step 1. Let x_2 be the largest x not covered by the relevant failure functions. x_2^2 + 1 is prime; let this be p_2. Repeat the iteration considering the interval x_2, x_2 + p_2. Repeat iterations outlined in iteration 1 and 2. Let p_i be the prime so obtained in iteration no.i. i.e. x_i^2 +1 =p_i. Note as i increases p_i also increases. Conjecture: after repeated iterations it will be found that the percentage of x not covered by the relevant failure functions is asymptotic to 3. To go from iteration i to iteration i + 1 we need only one x not covered by any of the relevant failure functions. The iteration comes to an end only if all the values of x in the interval x_i, x_i + p_i are covered by the failure functions. This is highly improbable. Since the iteration is perpetual there are infinitely many primes of the form x^2 + 1. Needless to say only a programmer can say, after repeated iterations, how alpha, the percentage of x not covered by the relevant failure functions becomes asymptotic to 3 or not. Conjecture: iteration is perpetual - hence there are infinitely many primes of form x^2 + 1.

[P] Application of failure functions - indirect primality test by akdevaraj Jul 1
Consider integers of the form x^2 + 1. The relevant failure functions pertaining to f(x) = x^2 + 1 are x = 1 +2*k, 2 +5*k, 4 + 17k etc. All the values of x not covered by the above are such that f(x) are prime and these need not be tested for primality. Here k belongs to Z.

[P] Some results pertaining to Z(i).(contd) by akdevaraj Jun 26
In the case of 3-factor composites Euler's generalisation of Fermat's theorem works in the ring of Gaussian integers irrespective of the shape of the prime factors.

[P] Some results pertaining to Z(i). by akdevaraj Jun 25
a) Fermat's theorem works only in the case of primes of shape 4m+1. b)Euler's generalisation of Fermat's theorem works only when composite numbers each of which is prime of shape 4m + 1 ( in the case of two-member composites -the only exception being 15 ). (to be continued ).

[P] Euler's generalisation of Fermat's theorem ....... by akdevaraj Jun 23
This theorem states that if a^x + c = m then a^(x+k*phi(m)) +c is congruent to 0 (mod m). Here a,x and c belong to N, x is not fixed. k also belongs to N. Ref: ISSN 1550 - 3747

[P] Search engine by akdevaraj Jun 22
Search engine is still not functioning.

[P] Messages by akdevaraj Jun 22
I am not able to post messages; Unlord should do something about this.

[P] Euler's generalisation of Fermat's theorem ....... by akdevaraj Jun 21
Euler's generalisation of Fermat's theorem - a further generalisation -- this is the title of a paper presented at the Hawaii Internation Conference in 2004. This theorem works in the ring of Gaussian integers also. Ref: ISSN # 1550-3747

[P] Euler's generalisation of Fermat's theorem ....... by akdevaraj Jun 20
Ref: ISSN # 1550 - 3747 If any member is interested I can give further details.

[P] Euler's generalisation of Fermat's theorem .......(contd) by akdevaraj Jun 19
Ref: ISSN # 1550- 3747. In Z the theorem states that if a^n + c = m then a^(n +k* phi(m) +c is congruent to 0 mod(m). Here n and k belong to N. In Z(i) this is also true, phi(m) being only Eulerphi of the real part of m.

[P] Euler's generalisation of Fermat's theorem ....... by akdevaraj Jun 18
I had submitted a paper at the Hawaii International conference on mathematics entitled " Euler's generalisation of Fermat's theorem - a further generalisation in 2004 ". That paper pertained to the ring of integers. I now find that it is true in the ring of Gaussian integers too.

[P] Euler's generalisation of Fermat's theorem - a further gene by akdevaraj Jun 16
"Euler's generalisation of Fermat's theorem - a further generalisation" is the title of a paper presented at Hawaii International conference on Mathematics in 2004. The theorem is true in the ring of Gaussian integers too. Ref: ISSN # 1550 - 3747

[P] Euler's generalisation of Fermat's theorem - a further gen by akdevaraj Jun 16
Ref: ISSN # 1550 - 3747

Measure things by SKungen Jun 13
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