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The average number of rational points on genus two curves is bounded
We prove that, when genus two curves $C/\mathbb{Q}$ with a marked Weierstass point are ordered by height, the average number of rational points $\#|C(\mathbb{Q})|$ is bounded. The argument followsExpand
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Low-Lying Zeros of Maass Form L-Functions
The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions ofExpand
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The average number of integral points on elliptic curves is bounded
We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at mostExpand
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Decidability and Shortest Strings in Formal Languages
TLDR
We consider the problem of determining if L is nonempty. Expand
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Square-root cancellation for the signs of Latin squares
TLDR
We prove the Alon-Tarsi conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically. Expand
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Self-conjugate core partitions and modular forms
A recent paper by Hanusa and Nath states many conjectures in the study of self-conjugate core partitions. We prove all but two of these conjectures asymptotically by number-theoretic means. We alsoExpand
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Proof of a conjecture of Stanley-Zanello
  • Levent Alpoge
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 11 August 2013
TLDR
We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b. Expand
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van der Waerden and the Primes
  • Levent Alpoge
  • Mathematics, Computer Science
  • Am. Math. Mon.
  • 1 October 2015
TLDR
We prove the infinitude of the primes via an application of van der Waerden's theorem. Expand
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The second moment of the number of integral points on elliptic curves is bounded
In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219Expand
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