Elementary Number Theory Problems. Part II

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their infinitude; establishment of a kind of similarity between natural numbers and numbers that appear in an arithmetic progression, similar formulae for primes and the so called generalized twin primes in an arithmetic progression and their infinitude; generalization of Bertrand postulate and a Bertrand like postulate for twin primes; some elementary implications of a simple primality test, the use of Chinese remainder theorem in a possible proof of the Goldbach conjecture; Schinzel Sierpinski conjecture; and lastly the Mersenne primes and composites, Fermat primes, and their infinitude.

ISBN ().444.()()()71·2 250 Problems, in Elementary Number Theory .-WACLAW SIERPINSKI "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathematics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. There is, in addition, a section of miscellaneous problems. Included are problems on several levels of difficulty-some are relatively easy, others rather complex, and a number so abstruse that they originally were the subject of scientific research and their solutions are of comparatively recent date. All of the solutions are given thoroughly and in detail; they contain information on possible generalizations of the given problem and further indicate unsolved problems associated with the given problem and solution.

Preprints, 2023

Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p • n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p • n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 • n • p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 • n • p + p formula we put forth with another known formula.

Theoretical Mathematics & Applications, 2022

A Fermat number is a number of the form F n = 2 2 n + 1, where n is an integer ≥ 0. In this paper, we show [via elementary arithmetic congruences] the following two results (R.) and (R'.). (R.): For every integer n ≥ 3, F n − 1 ≡ 1 mod[j], where j ∈ {3, 5, 17}. (R'.): For every integer n > 0 such that n ≡ 2 mod[6], we have F n − 1 ≡ 16 mod[19]. Result (R.) immediately implies that for every integer d ≥ 0, there exists at most two primes of the form 2F n + 1 + 10d [in particular, for every integer d ≥ 0, the numbers of the form 2F n + 1 + 10d (where n ≥ 2) are all composites ]; result (R.) also implies that there are infinitely many composite numbers of the form 2 n +F n and for every r ∈ {−2, 16}, there exists only one prime of the form r + F n. Result (R'.) immediately implies that there are infinitely many composite numbers of the form 2 + F n. That being said, we use the result (R.) and a special case of a Theorem of Dirichlet on arithmetic progression to explain why it is natural to conjecture that for every r ∈ {0, 2}, there are infinitely many primes of the form r + F n .

Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and geometric progressions are exposed.

ISBN ().444.()()()71·2 250 Problems, in Elementary Number Theory .-WACLAW SIERPINSKI "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathematics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. There is, in addition, a section of miscellaneous problems. Included are problems on several levels of difficulty-some are relatively easy, others rather complex, and a number so abstruse that they originally were the subject of scientific research and their solutions are of comparatively recent date. All of the solutions are given thoroughly and in detail; they contain information on possible generalizations of the given problem and further indicate unsolved problems associated with the given problem and solution.

In this paper we collected problems, which was either proposed or follow directly from results in our papers.

2013

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

iii Certificate This is to certify that the Seminar entitled "Number Theory " submitted by Sachin Solanki(10BIT049), towards the partial fulfillment of the requirements for the degree of Bachelor of Technology in Information Technology of Nirma University, Ahmedabad is the record of work carried out by him under my supervision and guidance.