[P] **Application of failure functions- sketch proof.** by akdevaraj Jul 5The 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 1Consider 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 26In 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 25a) 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 23This 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 22Search engine is still not functioning.

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

[P] **Euler's generalisation of Fermat's theorem .......** by akdevaraj Jun 21Euler'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 20Ref: 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 19Ref: 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 18I 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 16Ref: ISSN # 1550 - 3747

**Measure things** by SKungen Jun 13>