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primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> m*prod(k=1,pi(sqrt(m)),1 - 1/prime(k)) + n - 1 = pi(m) ... [1]

>

> where pi(m) is the number of primes not exceeding m.

> Clearly, every solution to [1] is rational.

> The largest such rational number that I have found is

>

> {m =

> 47207235513753656979626880173334817417470530133964196593860667/

> 1171662372654492721849024419009716384297071411200000000000;}

It is rather easy to prove that the number of solutions to [1]

is finite, in direct contradiction to a conjecture by Sergey.

Consider the ratio

R(m) = (m*prod(k=1,pi(sqrt(m)),1 - 1/prime(k)) + n - 1)/pi(m)

of the left and right hand sides of [1]. Then as m tends to infinity,

we know, from the prime number theorem and Mertens' formula,

that R(m) tends to

R(infinity) = 2*exp(-Euler) =

1.1229189671337703396482864295817615735314207738503063363083...

which is not equal to unity. Hence [1] has a finite number of solutions.

David