Generalized Fermat equations: A miscellany

Bennett, Michael A.; Chen, Imin; Dahmen, Sander R.; Yazdani, Soroosh

doi:10.1142/s179304211530001x

Cited by 31 publications

(46 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It states that the only non-trivial primitive solutions to (1) with χ < 1 are Bennett (2006); (2, 6, p) by Bennett and Chen (2012); and other signatures by other researchers. An excellent, exhaustive and up-to-date survey was recently compiled by Bennett, Chen, Dahmen, and Yazdani (2015a), which also proves the generalized Fermat conjecture for several families of signatures, including (2p, 4, 3).…”

Section: Introductionmentioning

confidence: 89%

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Anni¹,

Siksek²

2016

Algebra Number Theory

View full text Add to dashboard Cite

Abstract. Let K be a real abelian field of odd class number in which 5 is unramified. Let S 5 be the set of places of K above 5. Suppose for every non-empty proper subset S ⊂ S 5 there is a totally positive unit u ∈ O K such that q∈S Norm Fq/F 5 (u mod q) = 1. We prove that every semistable elliptic curve over K is modular, using a combination of several powerful modularity theorems and class field theory. We deduce that if K is a real abelian field of conductor n < 100, with 5 ∤ n and n = 29, 87, 89, then every semistable elliptic curve E over K is modular.Let ℓ, m, p be prime, with ℓ, m ≥ 5 and p ≥ 3. To a putative non-trivial primitive solution of the generalized Fermat x 2ℓ +y 2m = z p we associate a Frey elliptic curve defined over Q(ζp) + , and study its mod ℓ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if p ≤ 11, or if p = 13 and ℓ, m = 7.

show abstract

Section: Introductionmentioning

confidence: 89%

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Anni¹,

Siksek²

2016

Algebra Number Theory

View full text Add to dashboard Cite

show abstract

“…Fermat's Last Theorem established that A n + B n = C n has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C co-prime which would mean they do not have a common prime factor [14]. Hence, Fermat's Last Theorem can be seen as a special case of the Beal's conjecture restricted to x = y = z [4].…”

Section: Discussionmentioning

confidence: 99%

“…Billionaire banker Andrew Beal claims to have discovered this conjecture in 1993 while investigating generalizations of Fermat's Last Theorem [17]. This conjecture has occasionally been referred to as a generalized Fermat equation [4] and the Mauldin or Tijdeman-Zagier conjecture [11].…”

Section: Beal's Conjecturementioning

confidence: 99%

The Complexity of Number Theory

Vega¹

2020

Preprint

View full text Add to dashboard Cite

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem.

show abstract

“…It follows that for eachP ∈ C l (F 3 ) there is at most one P ∈ C l (Q) that reduces toP . Now the rational points ∞, (1,5) and (1, −5) respectively reduce to ∞, (1, 2), (1,1). To complete the proof it is sufficient to show that no Q-rational point reduces to (2, 0).…”

Section: Proof Of Theoremmentioning

confidence: 93%

Perfect powers expressible as sums of two fifth or seventh powers

Dahmen

Siksek

2014

Acta Arith.

Self Cite

View full text Add to dashboard Cite

Abstract. We show that the generalized Fermat equations with signatures (5, 5, 7), (5,5,19), and (7, 7, 5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive integer solutions for the signatures (5, 5, 11), (5,5,13), and (7, 7, 11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.

show abstract

Generalized Fermat equations: A miscellany

Cited by 31 publications

References 56 publications

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+ y2m= zp

The Complexity of Number Theory

Perfect powers expressible as sums of two fifth or seventh powers

Contact Info

Product

Resources

About