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A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4 k 2 .

Let q 1 = 2. Supposing that we have defined q j for all 1 j k, let q k+1 be a prime factor of 1 + Q k j=1 q j. As was shown by Euclid over two thousand years ago, q 1 , q 2 , q 3 ,. .. is then an infinite sequence of distinct primes. The sequence {q i } is not unique, since there is flexibility in the choice of the prime q k+1 dividing 1 + Q k j=1 q j.… (More)

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. In 1913, Dickson showed that for each fixed k, there are only finitely many odd perfect numbers with at most k distinct prime factors. We show how this result, and many like it, follow from embedding the natural… (More)

- Kevin Ford, Paul Pollack
- 2012

For each positive-integer valued arithmetic function f , let V f ⊂ N denote the image of f , and put V f (x) := V f ∩ [1, x] and V f (x) := #V f (x). Recently Ford, Luca, and Pomerance showed that V φ ∩ V σ is infinite, where φ denotes Euler's totient function and σ is the usual sum-of-divisors function. Work of Ford shows that V φ (x) ≍ V σ (x) as x → ∞.… (More)

- Paul Pollack
- 2016

Fix a number field K. For each nonzero α ∈ Z K , let ν(α) denote the number of distinct, nonassociate irreducible divisors of α. We show that ν(α) is normally distributed with mean proportional to (log log |N (α)|) D and standard deviation proportional to (log log |N (α)|) D−1/2. Here D, as well as the constants of proportionality, depend only on the class… (More)

- Saleh al-Alaiyan, Paul Pollack, Gerard F Notario
- Drugs in context
- 2015

BACKGROUND
The peak season of respiratory syncytial virus (RSV) infections in warmer climates may extend beyond the typical five-month RSV season of temperate regions. Additional monthly doses of palivizumab may be necessary in warmer regions to protect children at high risk for serious infection by the RSV.
METHODS
In a Phase II, single-arm,… (More)

- Benjamin Linowitz, Jeffrey S Meyer, Paul Pollack
- 2015

We examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally ge-odesic surfaces. Using techniques from analytic number theory, we address the following… (More)

It is well-known that Euclid's argument can be adapted to prove the infinitude of primes of the form 4k − 1. We describe a simple proof that the sum of the reciprocals of all such primes diverges. More generally, if q is a positive integer, and H is a proper subgroup of the units group (Z/qZ) × , we show that p prime p mod q ∈H 1 p = ∞.