[7] **viXra:1102.0058 [pdf]**
*submitted on 28 Feb 2011*

**Authors:** Tim Jones

**Comments:**
17 pages.

Transcendence of a number implies the irrationality of powers of a
number, but in the case of π there are no separate proofs that powers
of π are irrational. We investigate this curiosity. Transcendence proofs
for e involve what we call Hermite's technique; for π's transcendence
Lindemann's adaptation of Hermite's technique is used. Hermite's
technique is presented and its usage is demonstrated with irrationality
proofs of e and powers of e. Applying Lindemann's adaptation
to a complex polynomial, π is shown to be irrational. This use of
a complex polynomial generalizes and powers of π are shown to be
irrational. The complex polynomials used involve roots of i and yield
regular polygons in the complex plane. One can use graphs of these
polygons to visualize various mechanisms used to proof π^{2}, π^{3}, and π^{4}
are irrational. The transcendence of π and e are easy generalizations
from these irrational cases.

**Category:** Number Theory

[6] **viXra:1102.0051 [pdf]**
*replaced on 8 Apr 2011*

**Authors:** Chun-Xuan Jiang

**Comments:**
9 pages.

The Diophantine equations a^{2} ± m b^{2} = c^{n} ,
and a^{3} ± m b^{3} = d^{2} have infinitely many nonzero
integer solutions, Using the methods of infinite descent and infinite ascent we prove
y_{1}^{4} ± m y_{2}^{4} = R^{2} .

**Category:** Number Theory

[5] **viXra:1102.0046 [pdf]**
*submitted on 25 Feb 2011*

**Authors:** Chun-Xuan Jiang

**Comments:**
16 pages.

D.Zagier(1984) and K.Inkeri(1990) said[7] Jiang mathematics is true, but Jiang
determinates the irrational numbers to be very difficult for prime exponent p>2.In 1991
Jiang studies the composite exponents n=15,21,33,...,3p and proves Fermat last theorem for
prime exponent p>3[1].In 1986 Gerhard Frey places Fermat last theorem at elliptic
curve ,now called a Frey curve.Andrew Wiles studies Frey curve.In 1994 Wiles proves
Fermat last theorem[9,10].Conclusion:Jiang proof is direct and very simple,but Wiles proof
is indirect and very complex. If China mathematicians and Academia Sinica had supported
and recognized Jiang proof on Fermat last theorem,Wiles would not have proved Fermat
last theorem,because in 1991 Jiang had proved Fermat last theorem[1].Wiles has received
many prizes and awards, he should thank China mathematicians and Academia Sinica.To
support and to publish Jiang Fermat last theorem paper is prohibited in Academia Sinica.
Remark. Chun-Xuan Jiang,A general proof of Fermat last theorem(Chinese),Mimeograph
papers,July 1978. In this paper using circulant matrix,circulant determinant and
permutation group theory Jiang had proved Fermat last theorem for odd prime exponent.

**Category:** Number Theory

[4] **viXra:1102.0024 [pdf]**
*submitted on 15 Feb 2011*

**Authors:** Aziz Sahraei

**Comments:**
7 pages.

Always, when viewing papers whose writers show polyominoes graphically, this question crossed my mind, are there
any equations which may be given to avoid the need for drawings? Polyominoes are sometimes called by the number
of faces (like triomeno or tetraomino). In this paper, I try to formulate polyomino shapes and establish a
correspondence between them and polynominals. About the final part where I refer to the Goldbach conjecture,
I must to say that my aim is to give a geometric representation of the proof of this conjecture so that if a
special chain of subsets such as, (see paper) exists in a set Ω, where both ends of the chain include trivial
subsets, and if the conjecture be true for at least one arbitrary member of this chain, then it will be true
for all the other members of the chain.

**Category:** Number Theory

[3] **viXra:1102.0017 [pdf]**
*submitted on 11 Feb 2011*

**Authors:** Chun-Xuan Jiang

**Comments:**
20 pages.

1637 Fermat @@wrote: "It is impossible to separate a cube into two cubes, or a biquadrate
into two biquadrates, or in general any power higher than the second into powers of like degree: I
have discovered a truly marvelous proof, which this margin is too small to contain." part (4)

**Category:** Number Theory

[2] **viXra:1102.0011 [pdf]**
*submitted on 8 Feb 2011*

**Authors:** Subhajit Ganguly

**Comments:**
5 pages.

The following paper deals with the distribution of prime numbers, the twin prime
numbers and the Goldbach conjecture. Starting from the simple assertion that prime
numbers are never even, a rule for the distribution of primes is arrived at. Following the
same approach, the twin prime conjecture and the Goldbach conjecture are found to be
true.

**Category:** Number Theory

[1] **viXra:1102.0008 [pdf]**
*submitted on 7 Feb 2011*

**Authors:** Chun-Xuan Jiang

**Comments:** 18 pages.

D.Zagier(1984) and K.Inkeri(1990) said[7]:Jiang mathematics is true,but Jiang
determinates the irrational numbers to be very difficult for prime exponent p.In
1991 Jiang studies the composite exponents n=15,21,33,...,3p and proves Fermat
last theorem for prime exponenet p>3[1].In 1986 Gerhard Frey places Fermat last
theorem at the elliptic curve that is Frey curve.Andrew Wiles studies Frey curve.
In 1994 Wiles proves Fermat last theorem[9,10]. Conclusion:Jiang proof(1991) is
direct and simple,but Wiles proof(1994) is indirect and complex.If China
mathematicians had supported and recognized Jiang proof on Fermat last
theorem,Wiles would not have proved Fermat last theorem,because in 1991 Jiang
had proved Fermat last theorem.Wiles has received many prizes and awards,he
should thank China mathematicians.

**Category:** Number Theory