More Like this

1. Prime Labelings of Snake Graphs

   Bigham, A.; Donovan, E.A.; Pack, J.; Turley, J; Wiglesworth, L.W. (August 2019, PUMP journal of undergraduate research)

   null (Ed.)

   A prime labeling of a graph G with n vertices is a labeling of the vertices with distinct integers from the set {1,2,...,n} such that the labels of any two adjacent vertices are relatively prime. In this paper, we introduce a snake graph, the fused union of identical cycles, and define a consecutive snake prime labeling for this new family of graphs. We characterize some snake graphs that have a consecutive snake prime labeling and then consider a variation of this labeling. 

   more » « less
2. Poisson Approximation of Prime Divisors of Shifted Primes

   https://doi.org/10.1093/imrn/rnaf079  

   Ford, Kevin (April 2025, International Mathematics Research Notices)

   Abstract We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers $$\leqslant x$$ and of random shifted primes p+a with $$p\leqslant x$$. 

   more » « less
3. Partitio Numerorum: sums of a prime and a number of $$k$$-th powers

   https://doi.org/10.4171/jems/1567  

   Brüdern, Jörg; Wooley, Trevor D (November 2024, Journal of the European Mathematical Society)

   Let $$k$$ be a natural number and let $$c=2.134693\ldots$$ be the unique real solution of the equation $$2c=2+\log (5c-1)$$ in $$[1,\infty)$$. Then, when $$s\ge ck+4$$, we establish an asymptotic lower bound of the expected order of magnitude for the number of representations of a large positive integer as the sum of one prime and $$s$$ positive integral $$k$$-th powers. 

   more » « less
4. Joint Poisson distribution of prime factors in sets

   https://doi.org/10.1017/S0305004121000499  

   FORD, KEVIN (July 2022, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstarct Given disjoint subsets T 1 , …, T m of “not too large” primes up to x , we establish that for a random integer n drawn from [1, x ], the m -dimensional vector enumerating the number of prime factors of n from T 1 , …, T m converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T 1 , …, T m are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors. 

   more » « less
5. Almost-prime times in horospherical flows on the space of lattices

   https://doi.org/10.3934/jmd.2019022  

   McAdam, Taylor (November 2019, Journal of Modern Dynamics)

   An integer is called almost-prime if it has fewer than a fixed number of prime factors. In this paper, we study the asymptotic distribution of almost-prime entries in horospherical flows on Gamma\SL(n,R), where Gamma is either SL(n,Z) or a cocompact lattice. In the cocompact case, we obtain a result that implies density for almost-primes in horospherical flows where the number of prime factors is independent of basepoint, and in the space of lattices we show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. Along the way we give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, as well as an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows. 

   more » « less