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タイトル Primary pseudoperfect numbers, arithmetic progressions, and the Erd\H{o}s-Moser equation 2014年10月20日 16:30-17:30 Jonathan Sondow 氏 矢上キャンパス14棟204教室 A primary pseudoperfect number (PPN for short) is an integer $K > 1$ such that $$1/K + \sum_{p | K} 1/p = 1$$ where $p$ denotes a prime. I review what is known about PPNs, mentioning connections with perfectly weighted graphs, singularities of algebraic surfaces, Giuga numbers, Znam's problem, Sylvester's sequence, and Curtiss's bound on solutions to a unit fraction equation. I first show that if 6 divides $K$, then $K \equiv 6 \pmod{6^2}$. Reducing the known PPNs modulo $6^2 \times 8$, I then uncover a remarkable 7-term arithmetic progression. On that basis, I pose a conjecture which leads conditionally to a new record lower bound on any non-trivial solution to the Erd\H{o}s-Moser equation $$1^n + 2^n + . . . +(k-1)^n+ k^n = (k+1)^n.$$ The proof uses the Carlitz-von Staudt theorem and estimates of the error term in Mertens's theorem on prime harmonic sums. This is joint work with Kieren MacMillan.