Number Theory

1102 Submissions

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

The Powers of π are Irrational

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 π², π³, and π⁴ 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

The Diophantine Equations A² ± M B² = Cⁿ, A³ ± M B³ = D² and Y₁⁴ ± M Y₂⁴ = R²

Authors: Chun-Xuan Jiang
Comments: 9 pages.

The Diophantine equations a² ± m b² = cⁿ , and a³ ± m b³ = d² have infinitely many nonzero integer solutions, Using the methods of infinite descent and infinite ascent we prove y₁⁴ ± m y₂⁴ = R² .
Category: Number Theory

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

Jiang and Wiles Who Has First Proved Fermat Last Theorem (1)

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

An Analytical Approach to Polyominoes and a Solution to the Goldbach Conjecture

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

Jiang and Wiles Proofs on Fermat Last Theorem (4)

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

Distribution of Prime Numbers,Twin Primes and Goldbach Conjecture

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

Jiang And Wiles Proofs On Fermat Last Theorem (3)

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