Euler's totient function maximum value for a range

Question

For the euler's totient function, we have a number $n<10^{18}$

we have to find the value of $i$ between $2$ and $n$ (both inclusive) such that the value of $\phi(i)/i$ is maximum.

I have have observed that this value will be equal to the largest prime number less than or equal to n. Now since $n$ is upto $10^{18}$, what will be the most efficient way to do this?

http://math.stackexchange.com/questions/381053/maximum-of-frac-phiii — lab bhattacharjee, May 05 '13 at 08:01
http://math.stackexchange.com/questions/380440/largest-prime-below-a-given-number-n/380449#380449 — Daniel McLaury, May 05 '13 at 08:07
Since you already reduced to a question about looking for prime numbers, your title could have said so more clearly. — Marc van Leeuwen, May 05 '13 at 08:50

score 1 · Accepted Answer · answered May 05 '13 at 08:17

1

Hint:

$\phi(p)=p-1$, so now you have to look for largest prime before $10^{18}$

answered May 05 '13 at 08:17

Inceptio

7,881

1

din't i mention that above already? – Salena May 05 '13 at 08:18
Write a program to find out a largest prime before 10^{18}. – Inceptio May 05 '13 at 08:28
Even for $i=p^m$, $\phi(i)/i=(p-1)/p$, so need to look for primes and prime powers! – Karthik C May 05 '13 at 08:31
@AneeshKarthikC: When $i=p^m$, $\phi(i)=p^m(\dfrac{p-1}{p})$, right? – Inceptio May 05 '13 at 08:34
$\phi(i)/i$ I remarked about – Karthik C May 05 '13 at 08:42
Very fine. I missed the fact. Btw, is it pretty good in CMI? – Inceptio May 05 '13 at 08:45
@AneeshKarthikC: The value of $\phi(i)/i$ for $i=p^m$ does not exceed the value for $i=p$, so there is no need to separately test prime powers. Indeed as there are always other primes between $p$ and $p^m$ for $m>1$, a strict prime power will never maximise $\phi(i)/i$ in the range. – Marc van Leeuwen May 05 '13 at 08:46
What's pretty good? Number Theory? What exactly do you ask? – Karthik C May 05 '13 at 08:47
@AneeshKarthikC: I'm asking about the Chennai Math Institute. – Inceptio May 05 '13 at 08:47
@Marc: I stand corrected. – Karthik C May 05 '13 at 08:47

Euler's totient function maximum value for a range

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