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Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof [article]

Romeo Meštrović
2018 arXiv   pre-print
In Section 4, we give a new simple "Euclidean's proof" of the infinitude of primes.  ...  Furthermore, here are listed numerous elementary proofs of the infinitude of primes in different arithmetic progressions.  ...  Furstenberg's topological proof of IP and its modifications. A proof of Euclid's theorem due to H.  ... 
arXiv:1202.3670v3 fatcat:wbxwh5menzbixmhidj7777b6wa

On Formally Measuring and Eliminating Extraneous Notions in Proofs

A. Arana
2008 Philosophia Mathematica  
I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.  ...  Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved.  ...  Let Γ consist of formulas stating the premises of Furstenberg's topological proof, and let ∆ consist of a formula stating the infinitude of primes.  ... 
doi:10.1093/philmat/nkn015 fatcat:w4kto4hfmjbplnlkkt5aaf6sve

New Results on Primes from an Old Proof of Euler's [article]

Charles W. Neville
2003 arXiv   pre-print
In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers.  ...  Our short paper uses a simple modification of Euler's argument to obtain new results about the distribution of prime factors of sets of integers, including a weak one-sided Tschebyshev inequality.  ...  Furstenberg's proof is also provides an important beginning example in the theory of profinite groups [11, p. 476] Gotchev recently extended Furstenberg's work and gave a topological proof that P (S)  ... 
arXiv:math/0210282v2 fatcat:borjs4x3ivc7vp2zfjhdoxgznq

The Euclidean criterion for irreducibles [article]

Pete L. Clark
2016 arXiv   pre-print
We make connections with Furstenberg's "topological" proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products  ...  We recast Euclid's proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms.  ...  The essential core of his argument is the observation that in Z the set of elements not divisible by any prime number is ±1.  ... 
arXiv:1605.01298v1 fatcat:z3kbxhpeh5eajecdxfrw4nxg2y

Furstenberg and Margulis Awarded 2020 Abel Prize

Elaine Kehoe
2020 Notices of the American Mathematical Society  
"Note on One Type of Indeterminate Form" (1953) and "On the Infinitude of Primes" (1955) both appeared in the American Mathematical Monthly, the latter giving a topological proof of Euclid's famous theorem  ...  Furstenberg's proof was more conceptual than Szemerédi's, and it completely changed the area.  ... 
doi:10.1090/noti2110 fatcat:sgwngh4djrcgfax7ia2uzjyrhy

Teaching Interactive Proofs to Mathematicians

Mauricio Ayala-Rincón, Thaynara Arielly de Lima
2020 Electronic Proceedings in Theoretical Computer Science  
In particular, it is discussed how, using as case-of-study algebraic notions and properties, the use of the proof assistant Prototype Verification System PVS is promoted to interest mathematicians in the  ...  This work discusses an approach to teach to mathematicians the importance and effectiveness of the application of Interactive Theorem Proving tools in their specific fields of interest.  ...  The main interest of this position paper is to show how ITPs can be promoted among users of related areas, who do not necessarily require or desire to develop a strong background in proof theory and mathematical  ... 
doi:10.4204/eptcs.328.1 fatcat:fjnosq4nbfhvrhnc4bsd3hs6im

Finiteness Theorems for Perfect Numbers and Their Kin

Paul Pollack
2012 The American mathematical monthly  
We show how this result, and many like it, follow from embedding the natural numbers in the supernatural numbers and imposing an appropriate topology on the latter; the notion of sequential compactness  ...  Since ancient times, a natural number has been called perfect if it equals the sum of its proper divisors; e.g., 6 = 1+2+3 is a perfect number.  ...  Thanks are owed to Greg Martin, Michael Pollack, Carl Pomerance, Jonah Sinick, Enrique Treviño, Erick Wong, and the referees for helpful comments and conversations.  ... 
doi:10.4169/amer.math.monthly.119.08.670 fatcat:f4amm6jvhjcrhgqwo4v32sidea

Interview with Abel Laureate 2020 Hillel Furstenberg

Bjørn Ian Dundas, Christian Skau
2021 Notices of the American Mathematical Society  
Affi liate, Emeritus, and Nominee members are not eligible for this benefi t. R ec om m en d a fr ie nd Ea rn P oi nt s!  ...  We will focus on one of these papers titled "On the infinitude of primes." The paper is only half a page in length, but that belies its originality.  ...  So this idea of looking at something happening in the integers as taking place in a measure space came in a natural way from the early paper, "On the infinitude of primes."  ... 
doi:10.1090/noti2321 fatcat:pnonnalvvfbupbdkdelrhruxqm

Classifying Subatomic Domains [article]

Noah Lebowitz-Lockard
2016 arXiv   pre-print
In a recent paper of Boynton and Coykendall \cite{BC}, the two authors introduce two properties that are slightly weaker than atomicity, which they call "almost atomicity" and "quasiatomicity".  ...  In this paper, we classify various subatomic properties and show that they are all distinct.  ...  Acknowledgements This paper came out the University of Georgia 2015-2016 AGANT group. Funding support came from National Science Foundation RTG grant DMS-1344994.  ... 
arXiv:1610.05874v1 fatcat:waoewrdoirfrxegrl52rhjxlly

A Gauche perspective on row reduced echelon form and its uniqueness [article]

Eric L. Grinberg
2021 arXiv   pre-print
By means of the Gauche basis we interpret the row reduced echelon form of $M$, and give a direct proof of its uniqueness. We conclude with pedagogical reflections.  ...  Using a left-to-right "sweeping" algorithm, we define the \emph{Gauche basis} for the column space of a matrix $M$.  ...  Every such course covers Euclid's proof of the infinitude of primes, and rightly so. But we can also add H. Furstenberg's "topological" proof [4, 10] .  ... 
arXiv:2005.06275v3 fatcat:cnunkanxojgh3hiocxqnfaoey4

The book review column

William Gasarch
2006 ACM SIGACT News  
In this column we review the following books.  ...  Furstenberg's topological proof of the infinitude of the primes (Chapter 8) will likely be incomprehensible for many students, as will the last article, about Fermat's last theorem.  ...  The first two chapters in this section focus on the effect of the underlying network topology on the design and performance of matrix algorithms -chapter 4 discusses ring topologies, and chapter 5 focuses  ... 
doi:10.1145/1140612.1140614 fatcat:csh47xvp4jaapcebrtahj3u3jm

Some Fundamental Theorems in Mathematics [article]

Oliver Knill
2020 arXiv   pre-print
For non-linear sequences of numbers the problems are wide open. The Landau problem of the infinitude of primes of the form x 2 + 1 illustrates this.  ...  See [10] for proofs from the books, where the proofs have full details. Euclid: The set of primes is infinite. Proof: let p be largest prime, then p! + 1 has a larger prime factor than p.  ...  His argument is ingenious: assume, we could count the points a 1 , a 2 , . . . .  ... 
arXiv:1807.08416v3 fatcat:ijb3ei3yfbb6hckf7p5pvgscfm

Combinatorial and Additive Number Theory Problem Sessions: '09–'19 [article]

Steven J. Miller
2019 arXiv   pre-print
These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences).  ...  If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See for the conference homepage.  ...  Two types of proofs of the infinitude of primes, those that give lower bounds and those that don't (such as Furstenberg's topological proof). What category does ζ(2) = π 2 /6 = Q fall under?  ... 
arXiv:1406.3558v11 fatcat:fo54wybi4bg2tncpwfapgsjqb4

Recreational mathematics [chapter]

2009 Famous Puzzles of Great Mathematicians  
Fürstenberg's topological proof made easy There is a famous proof of the infinitude of primes using topology. It can be found in many books.  ...  C A B O P O 1 O 2 Q Chapter 21 Infinitude of prime numbers Proofs by construction of sequence of relatively prime numbers Fibonacci numbers 1 Since gcd(F m , F n ) = F gcd(m,n) , if there are only  ...  Find the sum of the first n odd numbers. 2. Find the sum of the cubes of the first n odd numbers. 3. Find S 5 (n) and S 6 (n). 4.  ... 
doi:10.1090/mbk/063/01 fatcat:w6hwtau7gvek3fqd4iv6ospscu

On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method

Elon Lindenstrauss, Manfred Einsiedler
2007 Journal of Modern Dynamics  
We consider measures on locally homogeneous spaces Γ\G which are invariant and have positive entropy with respect to the action of a single diagonalizable element a ∈ G by translations, and prove a rigidity  ...  This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [Lin2, EKL] is used to classify positive entropy measures invariant under a one parameter group with  ...  We thank Shahar Mozes for helpful comments; we also thank the anonymous referee for his timely and helpful referee report.  ... 
doi:10.3934/jmd.2008.2.83 fatcat:dzqq44izpvgyzfmjs4qf26zfti