Re: Pi(n) question

daniel t wrote :

A propos, Mathematica gives Pi[n]> Pi[kn] -
Pi[(k-1)n] for a rather large value of n, and
striking visual evidence that the conjecture is true,
notwithstanding that the difference jumps around a
lot.

I attribute the short computational time to the
pre-programming in Mathematica's PrimePi function. Is
there any reason to question this data?

The software doesn't hesitate until one reaches
10,000,000,000 (PrimePi=455 million or so), and then
it takes a second or so.

as usual i will give a different reply.

although note im not so convinced here , rather an impulsive guess.

for some reason , that is too complicated or long to include here , i will approximate Pi(x).

your conjecture becomes by the approximation x/log(x) ;

n/log(n) > k*n/(log(k)+log(n)) - (k-1)n/(log(k-1)+log(n))

divide all by n ; ( and rearrange )

1/log(n) - 1/(log(k-1) + log(n)) > k/(log(k) + log(n)) - k/(log(k-1) + log(n))

think about it.

so for n >> 100 and k >> n , what is your conclusion ?

hint log(k) - log(k-1) is very small for large k.

regards

tommy1729

" statisticly , i dont exist " tommy1729
.

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