Binary Egyptian Fractions

✍ Scribed by Ernest S Croot III; David E Dobbs; John B Friedlander; Andrew J Hetzel; Francesco Pappalardi

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 144 KB
• Volume: 84
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2508

✦ Synopsis

Let A k *(n) be the number of positive integers a coprime to n such that the equation aÂn=1Âm 1 + } } } +1Âm k admits a solution in positive integers (m 1 , ..., m k ). We prove that the sum of A 2 *(n) over n x is both > >x log 3 x and also < <x log 3 x. For the corresponding sum where the a's are counted with multiplicity of the number of solutions we obtain the asymptotic formula. We also show that A k *(n)< <n : k += where : k is defined recursively by : 2 =0 and : k =1&(1&: k&1 )Â (2+: k&1 ).