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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. 

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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 $$p\leqslant x$$. 

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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. 

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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. 

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5. CCAT-prime Collaboration: Science Goals and Forecasts with Prime-Cam on the Fred Young Submillimeter Telescope

   https://doi.org/10.3847/1538-4365/ac9838  

   CCAT-Prime Collaboration; Aravena, Manuel; Austermann, Jason E.; Basu, Kaustuv; Battaglia, Nicholas; Beringue, Benjamin; Bertoldi, Frank; Bigiel, Frank; Bond, J. Richard; Breysse, Patrick C.; et al (December 2022, The Astrophysical Journal Supplement Series)

   Abstract We present a detailed overview of the science goals and predictions for the Prime-Cam direct-detection camera–spectrometer being constructed by the CCAT-prime collaboration for dedicated use on the Fred Young Submillimeter Telescope (FYST). The FYST is a wide-field, 6 m aperture submillimeter telescope being built (first light in late 2023) by an international consortium of institutions led by Cornell University and sited at more than 5600 m on Cerro Chajnantor in northern Chile. Prime-Cam is one of two instruments planned for FYST and will provide unprecedented spectroscopic and broadband measurement capabilities to address important astrophysical questions ranging from Big Bang cosmology through reionization and the formation of the first galaxies to star formation within our own Milky Way. Prime-Cam on the FYST will have a mapping speed that is over 10 times greater than existing and near-term facilities for high-redshift science and broadband polarimetric imaging at frequencies above 300 GHz. We describe details of the science program enabled by this system and our preliminary survey strategies. 

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