Problem with divisors

Number Theory Level pending

In this problem d ( n ) d(n) means the numer of n n 's divisors.

d ( k + 1 ) ≥ 2 ∗ d ( k ) d(k+1)\geq 2*d(k)

How many k k postive integer are there, so the equation above hold true?

A) Infinite

B) More then 10 10 , but not infinite

C) 10 10

D) Lower than 10 10 , but higher than 5 5

E) 5 5

F) 3 3 or 4 4

G) 1 1 or 2 2

H) 0 0

H D F C G B A E

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Áron Bán-Szabó
Jun 11, 2017

Let k k be a prime, so that k = 4 m + 1 k=4m+1 , where m m is a positive integer. we will use that there are infinite primes, which make 1 1 remainder dividing by 4 4 . Then d ( k ) = 2 d(k)=2 and

d ( k + 1 ) = d ( 4 m + 2 ) = d ( 2 ( 2 m + 1 ) = d ( 2 ) ∗ d ( 2 m + 1 ) ≥ 2 ∗ 2 = 4 d(k+1)=d(4m+2)=d(2(2m+1)=d(2)*d(2m+1)\geq2*2=4 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...