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... Nonetheless, the availability of the code (or proof) can be tremendously helpful to verify or re-implement the method. It is indeed much easier to verify a result (with the initial code or proof), then it is to produce from nothing (this is perhaps most poignantly illustrated by the longevity of the lack of proof for Fermat's last theorem (Wikipedia, 2020).) ...

... There have been a great many erroneous proofs of FLT, see [67]. The list might well include Fermat's proof, referred to in his marginal note -we will never know with certainty, but in view of developments over the next 350 years, it seems very unlikely that his proof was correct. ...

We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructive errors that have been detected in the author's own published papers.

... Nonetheless, the availability of the code (or proof) can be tremendously helpful to verify or re-implement the method. It is indeed much easier to verify a result (with the initial code or proof), then it is to produce from nothing (this is perhaps most poignantly illustrated by the longevity of the lack of proof for Fermats last theorem (Wikipedia, 2020).) ...

One of the challenges in machine learning research is to ensure that presented and published results are sound and reliable. Reproducibility, that is obtaining similar results as presented in a paper or talk, using the same code and data (when available), is a necessary step to verify the reliability of research findings. Reproducibility is also an important step to promote open and accessible research, thereby allowing the scientific community to quickly integrate new findings and convert ideas to practice. Reproducibility also promotes the use of robust experimental workflows, which potentially reduce unintentional errors. In 2019, the Neural Information Processing Systems (NeurIPS) conference, the premier international conference for research in machine learning, introduced a reproducibility program, designed to improve the standards across the community for how we conduct, communicate, and evaluate machine learning research. The program contained three components: a code submission policy, a community-wide reproducibility challenge, and the inclusion of the Machine Learning Reproducibility checklist as part of the paper submission process. In this paper, we describe each of these components, how it was deployed, as well as what we were able to learn from this initiative.

... See[6] for the complete text. ...

Fermat's Last Theorem states that the Diophantine equation X^n + Y^n = Z^n has no non-trivial solution for any n greater than 2. In this paper we give an approach to a brief and simple proof of the theorem using only elementary methods.

... 5. There was a cash prize offered for its solution. To quote [81] Interest in FLT rocketed when a German doctor and amateur mathematician called Paul Wolfskehl offered a huge cash prize in 1908 for its solution. Many people, mostly amateur mathematicians, sent in their potential solutions to their nearest universities. ...

0.1 What is Ramsey Theory and why did we write this book?.. 7 0.2 What is a purely combinatorial proof?............. 8 0.3 Who could read this book?.................... 9 0.4 Abbreviations used in this book................. 9

... For an odd prime p not dividing xyz, A. Wieferich [43] showed that x p +y p +z p = 0 implies q p (2) ≡ 0 (mod p). The only known such primes (the so called Wieferich primes) 1093 and 3511 have long been known, and it was reported in [5] that there exist no new Wieferich primes p < 4×10 12 . ...

Let p > 3 be a prime, and let q
p
(2) = (2p−1 − 1)/p be the Fermat quotient of p to base 2. In this note we prove that $$\sum\limits_{k = 1}^{p - 1} {\frac{1}{{k \cdot {2^k}}}} \equiv {q_p}(2) - \frac{{p{q_p}{{(2)}^2}}}{2} + \frac{{{p^2}{q_p}{{(2)}^3}}}{3} - \frac{7}{{48}}{p^2}{B_{p - 3}}(\bmod {p^3})$$, which is a generalization of a congruence due to Z.H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z.H. Sun, we show that $${q_p}{(2)^3} \equiv - 3\sum\limits_{k = 1}^{p - 1} {\frac{{{2^k}}}{{{k^3}}}} + \frac{7}{{16}}\sum\limits_{k = 1}^{(p - 1)/2} {\frac{1}{{{k^3}}}} (\bmod p)$$, which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum \(\sum\limits_{k = 1}^{p - 1} {{1 \mathord{\left/
{\vphantom {1 {\left( {k^2 \cdot 2^k } \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {k^2 \cdot 2^k } \right)}}}\) modulo p
2 that also generalizes a related Sun’s congruence modulo p.

This preprint is all about Fermat last theorem using a simple arithmetic and the principles of digit number.
There are two ways to proof that the both sides of a Pythagorean formula with power n>2 are equal. One is by using simple arithmetic operations; second is by using digit sum number theorem. The article 'Fermat's last theorem: using simple arithmetic operation and the principle of digit sum number theorem' by Gbenga Ayodele, have shown simple mathematical ways to proof this.

For over 300 years Fermat's last theorem has been one of the most difficult problems to be solved in the field of mathematics. The statement says that “It is impossible for a
cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth
powers or, in general, for any number which is a power greater than the second to be written as a
sum of two like powers."
�Fermat claims to have a general solution to the problem, but left no proof behind except for the power n=4; using infinite descent which he developed during his time. Many geniuses in the field have contributed to solving the problem using the same approach, but left the proofs uncompleted. Andrew Wiles uses the same method with a more advance approach, his solution was accepted in 1993 and later published in 1995. The idea of a proof by infinite descent is simply by assuming that there is a positive integral solution to your problem. Then through algebraic manipulation, show that given this
solution you can find a smaller positive integral solution. This article make used of a digit sum number theorem and a simple arithmetic operations with Pythagorean triples to show a contradiction to Fermat's statement. Using the digit sum theorem both sides of the Pythagorean equation may not be equal in their real number but are equal in their simple digits. While checking the value for the hypoteneus z as the power n increases using simple arithmetic operation, x tends to zero and the hypotenuse z
tends to y”.� and at a certain power n, the both sides of the Pythagorean equation are equal.

This short summary was based on the number theory's application in physics where I mainly took an focus on quaternions and several topics in W.S. Anglin and J.Lambek's The Heritage of Thales.

Letlbe any odd prime andζanlth root of unity. LetCl(i)be the eigenspace of the ideal class group ofQ(ζ) corresponding toωi, ωbeing the Teichmuller character. We prove that if the order ofCl(3)islh3, thenxln+yln+zln=0 has no solutions withl∤xyz,n⩾max(1, h3).

tion often posed to An-drew Wiles in interviews, namely, what fasci-nated him so greatly in the Fermat conjec-ture, he seldom re-frained from answering by emphasizing the long history of this problem. When I asked him the same question in Boston in 1995, he answered, "Because of its romantic history. " When I then went further and asked him to explain to me in more detail what he meant by romantic, he an-swered merely, "Because Fermat said he had a

In earlier work, we introduced flexible inference and decision-theoretic
metareasoning to address the intractability of normative inference. Here,
rather than pursuing the task of computing beliefs and actions with decision
models composed of distinctions about uncertain events, we examine methods for
inferring beliefs about mathematical truth before an automated theorem prover
completes a proof. We employ a Bayesian analysis to update belief in truth,
given theorem-proving progress, and show how decision-theoretic methods can be
used to determine the value of continuing to deliberate versus taking immediate
action in time-critical situations.

In this paper we show how one can use an inner product formula of Heim giving the inner product of a pullback Eisenstein se-ries from Sp 10 to Sp 2 × Sp 4 × Sp 4 with a newform on GL2 and a Saito-Kurokawa lift to produce congruences between Saito-Kurokawa lifts and non-CAP forms. This congruence is in part controlled by the L-function on GSp 4 × GL2. The congruence is then used to produce nontrivial tor-sion elements in an appropriate Selmer group, providing evidence for the Bloch-Kato conjecture.

We consider the conceptual foundations of the renormalization-group (RG) formalism. We show that the RG map, defined on a suitable space of interactions, is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the other hand, we prove in several cases that near a first-order phase transition the renormalized measure is not a Gibbs measure for any reasonable interaction. It follows that the conventional RG description of first-order transitions is not universally valid.

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