Case Of Fermat Research Articles

Fermat's Last Theorem, Schur's Theorem (in Ramsey Theory), and the infinitude of the primes

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.

On Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors

Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat.

An extension of Legendre's criterion in connection with the first case of Fermat's last theorem

An extension of Legendre's criterion in connection with the first case of Fermat's last theorem

Correlation measures of binary sequences derived from Euler quotients

<abstract><p>Fermat-Euler quotients arose from the study of the first case of Fermat's Last Theorem, and have numerous applications in number theory. Recently they were studied from the cryptographic aspects by constructing many pseudorandom binary sequences, whose linear complexities and trace representations were calculated. In this work, we further study their correlation measures by introducing a new approach based on Dirichlet characters, Ramanujan sums and Gauss sums. Our results show that the $ 4 $-order correlation measures of these sequences are very large. Therefore they may not be suggested for cryptography.</p></abstract>

Remarques sur le premier cas du théorème de Fermat sur les corps de nombres

The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, t

A search for primes 𝑝 such that the Euler number 𝐸_{𝑝-3} is divisible by 𝑝

Let p > 3 p>3 be a prime. Euler numbers E p − 3 E_{p-3} first appeared in H. S. Vandiver’s work (1940) in connection with the first case of Fermat’s Last Theorem. Vandiver proved that if x p + y p = z p x^p+y^p=z^p has a solution for integers x , y , z x,y,z with gcd ( x y z , p ) = 1 \gcd (xyz,p)=1 , then it must be that E p − 3 ≡ 0 ( mod p ) E_{p-3}\equiv 0\,(\bmod \,p) . Numerous combinatorial congruences recently obtained by Z.-W. Sun and Z.-H. Sun involve the Euler numbers E p − 3 E_{p-3} . This gives a new significance to the primes p p for which E p − 3 ≡ 0 ( mod p ) E_{p-3}\equiv 0\,(\bmod \,p) . For the computation of residues of Euler numbers E p − 3 E_{p-3} modulo a prime p p , we use a congruence which runs significantly faster than other known congruences involving E p − 3 E_{p-3} . Applying this, congruence, via a computation in Mathematica 8, shows that there are only three primes less than 10 7 10^7 that satisfy the condition E p − 3 ≡ 0 ( mod p ) E_{p-3}\equiv 0\,(\bmod \,p) (these primes are 149, 241 and 2946901). By using related computational results and statistical considerations similar to those used for Wieferich, Fibonacci-Wieferich and Wolstenholme primes, we conjecture that there are infinitely many primes p p such that E p − 3 ≡ 0 ( mod p ) E_{p-3}\equiv 0\,(\bmod \,p) .

On the Applications of Cyclotomic Fields in Introductory Number Theory

In this essay, we study and comment on two number theoretical applications on prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime). We begin by giving a simplified proof of Kummer's case of Fermat's Last Theorem obtained by linking different versions of the proof in different textbooks. We finally modernize Dirichlet's solution to Pell's Equation.

Norm Residue Symbol and the First Case of Fermat's Equation

Norm Residue Symbol and the First Case of Fermat's Equation

Fermat's equation over the tower of cyclotomic fields

Let be a prime, let let be the maximal real subfield of , and let be the maximal -subextension of . We define effectively calculable integer-valued functions , and such that , where is the index of irregularity of . For we prove the first case of Fermat's theorem for , , and . We obtain explicit lower estimates for , and . For regular (when ) we prove the second case of Fermat's theorem for and and Fermat's theorem for , and , generalizing the classical result on the validity of Fermat's theorem for and regular . We also obtain some other results on solutions of Fermat's equation over , and .

On the first case of Fermat's theorem for cyclotomic fields

The classical criteria of Kummer, Mirimanov and Vandiver for the validity of the first case of Fermat's theorem for the field of rationals and prime exponent are generalized to the field and exponent . As a consequence, some simpler criteria are established. For example, the validity of the first case of Fermat's theorem is proved for the field and exponent on condition that does not divide .

A new criterion for the first case of Fermat’s last theorem

It is shown that if the first case of Fermat’s last theorem fails for an odd prime l, then the sums of reciprocals modulo l, s ( k , N ) = ∑ 1 / j ( k l / N > j > ( k + 1 ) l / N ) s(k,N) = \sum 1/j\;(kl/N > j > (k + 1)l/N) are congruent to zero mod l \bmod \;l for all integers N and k with 1 ≤ N ≤ 46 1 \leq N \leq 46 and 0 ≤ k ≤ N − 1 0 \leq k \leq N - 1 . This is equivalent to B l − 1 ( k / N ) − B l − 1 ≡ 0 ( mod l ) {B_{l - 1}}(k/N) - {B_{l - 1}} \equiv 0 \pmod l , where B n {B_n} and B n ( x ) {B_n}(x) are the nth Bernoulli number and polynomial, respectively. The work can be considered as a result on Kummer’s system of congruences.

A New Criterion for the First Case of Fermat's Last Theorem

A new criterion of the first case of Fermat Last Theorem is derived. This criterion is based on special sums of reciprocals.

A Special Extension of Wieferich's Criterion

The following theorem is proved in this paper: If the first case of Fermat's Last Theorem does not hold for sufficiently large prime 1, then E x1-2 kl < x < (k X I) =_ 0 (modl1) x NN for all pairs of positive integers N, k, N < 94, 0 < k < NI. The proof of this theorem is based on a recent paper of Skula and uses computer techniques. 0. INTRODUCTION The first case of Fermat's Last Theorem states that for each odd prime I the equation Xi +y1 + zi = 0 has no integral solution x, y, z with I t xyz. One of many methods investigating this problem was introduced by A. Wieferich. This method is connected with the Fermat quotients ql (a), al-l q, (a) = I defined for each integer a such that a is not divisible by 1. Let us assume in this paragraph that / is an odd prime which does not satisfy the first case of Fermat's Last Theorem. In 1909, Wieferich [7] published the following important result: q, (2) =0 (mod1) . Many mathematicians have extended this Wieferich criterion. The latest result is due to A. Granville and B. Monagan [1] and states q1 (p) _ 0 (mod 1) for each prime p such that p < 89. These considerations have been generalized by L. Skula. He studied the sums s(k, N) = E 1-2 (kl <x<(k + 1)1) s(k, N) = EX<X

On Krasnér's theorem for the first case of Fermat's last theorem

On Krasnér's theorem for the first case of Fermat's last theorem

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

Lower Bounds for the Solutions in the Second Case of Fermat's Last Theorem

Let p be an odd prime. In this paper, we prove that if p≢3 (mod 4) and x, y, z are integers satisfying x p +y p =z p , p/xyz, 0 2 −1/p p 6p−2 and z-x>1/2p 6p−3

Lower bounds for the solutions in the second case of Fermat’s last theorem

Let $p$ be an odd prime. In this paper, we prove that if $p \equiv 3$ and $x,y,z$ are integers satisfying ${x^p} + {y^p} = {z^p},p|xyz,0 < x < y < z$, then $y > {2^{ - 1/p}}{p^{6p - 2}}$ and $z - x > \tfrac {1}{2}{p^{6p - 3}}$.

The prime factors of Wendt’s binomial circulant determinant

Wendt’s binomial circulant determinant,Wm{W_m}, is the determinant of anmbymcirculant matrix of integers, with (i, j)th entry(m|i−j|)\left ( {\begin {array}{*{20}{c}} m \\ {|i - j|} \\ \end {array} } \right )whenever 2 dividesmbut 3 does not. We explain how we found the prime factors ofWm{W_m}for each evenm≤200m \leq 200by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that ifpandq=mp+1q = mp + 1are odd primes, 3 does not dividem, andm≤200m \leq 200, then the first case of Fermat’s Last Theorem is true for exponentp.

Fermat’s last theorem (case 1) and the Wieferich criterion

This note continues work by the Lehmers [3], Gunderson [2], Granville and Monagan [1], and Tanner and Wagstaff [6], producing lower bounds for the prime exponent p in any counterexample to the first case of Fermat’s Last Theorem. We improve the estimate of the number of residues r mod p 2 r\bmod {p^2} such that r p ≡ r mod p 2 {r^p} \equiv r\bmod {p^2} , and thereby improve the lower bound on p to 7.568 × 10 17 7.568 \times {10^{17}} .

New Bound for the First Case of Fermat's Last Theorem

New Bound for the First Case of Fermat's Last Theorem