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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]

2018
*
arXiv
*
pre-print

In Section 4, we give a new simple "Euclidean's

arXiv:1202.3670v3
fatcat:wbxwh5menzbixmhidj7777b6wa
*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. ...##
###
On Formally Measuring and Eliminating Extraneous Notions in Proofs

2008
*
Philosophia Mathematica
*

I argue that

doi:10.1093/philmat/nkn015
fatcat:w4kto4hfmjbplnlkkt5aaf6sve
*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*. ...##
###
New Results on Primes from an Old Proof of Euler's
[article]

2003
*
arXiv
*
pre-print

In 1737 Leonard Euler gave what we often now think

arXiv:math/0210282v2
fatcat:borjs4x3ivc7vp2zfjhdoxgznq
*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) ...##
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The Euclidean criterion for irreducibles
[article]

2016
*
arXiv
*
pre-print

We make connections with

arXiv:1605.01298v1
fatcat:z3kbxhpeh5eajecdxfrw4nxg2y
*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. ...##
###
Furstenberg and Margulis Awarded 2020 Abel Prize

2020
*
Notices of the American Mathematical Society
*

"Note on One Type

doi:10.1090/noti2110
fatcat:sgwngh4djrcgfax7ia2uzjyrhy
*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. ...##
###
Teaching Interactive Proofs to Mathematicians

2020
*
Electronic Proceedings in Theoretical Computer Science
*

In particular, it is discussed how, using as case-

doi:10.4204/eptcs.328.1
fatcat:fjnosq4nbfhvrhnc4bsd3hs6im
*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 ...##
###
Finiteness Theorems for Perfect Numbers and Their Kin

2012
*
The American mathematical monthly
*

We show how this result,

doi:10.4169/amer.math.monthly.119.08.670
fatcat:f4amm6jvhjcrhgqwo4v32sidea
*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. ...##
###
Interview with Abel Laureate 2020 Hillel Furstenberg

2021
*
Notices of the American Mathematical Society
*

Affi liate, Emeritus,

doi:10.1090/noti2321
fatcat:pnonnalvvfbupbdkdelrhruxqm
*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*." ...##
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Classifying Subatomic Domains
[article]

2016
*
arXiv
*
pre-print

In a recent paper

arXiv:1610.05874v1
fatcat:waoewrdoirfrxegrl52rhjxlly
*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. ...##
###
A Gauche perspective on row reduced echelon form and its uniqueness
[article]

2021
*
arXiv
*
pre-print

By means

arXiv:2005.06275v3
fatcat:cnunkanxojgh3hiocxqnfaoey4
*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] . ...##
###
The book review column

2006
*
ACM SIGACT News
*

In this column we review

doi:10.1145/1140612.1140614
fatcat:csh47xvp4jaapcebrtahj3u3jm
*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 ...##
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Some Fundamental Theorems in Mathematics
[article]

2020
*
arXiv
*
pre-print

For non-linear sequences

arXiv:1807.08416v3
fatcat:ijb3ei3yfbb6hckf7p5pvgscfm
*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 , . . . . ...##
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Combinatorial and Additive Number Theory Problem Sessions: '09–'19
[article]

2019
*
arXiv
*
pre-print

These notes are a summary

arXiv:1406.3558v11
fatcat:fo54wybi4bg2tncpwfapgsjqb4
*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 http://www.theoryofnumbers.com/ 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? ...##
###
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. ...

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

2007
*
Journal of Modern Dynamics
*

We consider measures on locally homogeneous spaces Γ\G which are invariant

doi:10.3934/jmd.2008.2.83
fatcat:dzqq44izpvgyzfmjs4qf26zfti
*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. ...