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タイトル Primary pseudoperfect numbers, arithmetic progressions, and the Erd\H{o}s-Moser equation
開催日時 2014年10月20日 16:30-17:30
主催者
講演者 Jonathan Sondow 氏
場所 矢上キャンパス14棟204教室
内容
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.
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