Let pk be the kth prime and suppose
u = p12∗p22∗p32...∗pm2=Πk=1k=mpk2
and
s = u
How many prime divisors would s have? How many prime divisors would 1.01 s have? How does this tell you how to pick an n?
Let pk be the kth prime and suppose
u = p12∗p22∗p32...∗pm2=Πk=1k=mpk2
and
s = u
How many prime divisors would s have? How many prime divisors would 1.01 s have? How does this tell you how to pick an n?
Sorry, what I should have said is how many divisors does n have, not how many prime divisors. Also, not that p1, p2, p3, ... are the 1st, 2nd, 3rd, ... primes but rather a set of primes such that p1 < p2 < p3 < ...
Now, more ramblings [I'm not sure we are going any where useful]. Let m=0 [i.e. n=1]. The number of divisors is 1=30.
[1]
Let m=1 [i.e. n = p12]. The number of divisors is 3 = 31
[1*[1], p1*[1], p12*[1]] = [1, p1, p12]
Let m=2. The number of divisors is 9=32
[1*[1, p1, p12], p2*[1, p1, p12], p22*[1, p1, p12]]
= [1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22]
Let m=3 The number of divisors is 27=33
[1*[1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22], p3*[1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22], p32*[1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22]]
=[...]
Now generalize that, i.e. s=n=Πk=1mpkq
The number of divisors of s is qm. If we let q=pk, then obviously d is greater than or equal to s. Can we find an m so that pkm< 1.01 s. What about m < log(1.01 s)/log(pk)
Is that enough to play around with for a while?
BTW: There should be some easier way to do this. Anybody?
Prove that there exists a natural number n, which has over 2017 divisors d satisfying: √n≤d<1,01√n
To be honest, I don't really know where to start, again, but I assume that: n=2a+3b+5c, where a+b+c>2016 (because 1 is also a divisor)
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