When should this question be answered?

Question

If you hover your mouse over the time marked in the right-bottom corner of this post, then you will see a string showing the exact time that I post this question, which reads 2021-03-18 22:32:47Z. See this post for more details on how to check the exact time of posting.

Ignoring all non-digit characters, that time gives the number $20210318223247$. It turns out that this number has exactly $4$ divisors.

Now I offer a bounty of 500 reputations to reward any answer to this question whose posting time gives a number that has exactly $5$ divisors.

When should you post your answer?

_{Note: If you post a correct answer at a time that does not meet the above criterion, the answer will still be accepted. You just don't get the bounty.}

Answer 1

I'm a very impatient person so I've decided not to wait around to post an answer at the right time which I think will be

7461-02-24 07:38:41Z

Reasoning

Numbers which have exactly five divisors are those of the form $p^4$ where $p$ is a prime so it is a matter of searching through fourth powers of primes which look like a date and time (some get quite close).

Answer 2

Having 5 divisors means

that the number is exactly the fourth power of a prime number.

A standard number-theoretical way to observe this is

to look at the properties of Number of Divisors function $\sigma_0(n)$, specifically $$ \sigma_0(p_1^{e_1}p_2^{e_2}\cdots) = (1+e_1)(1+e_2)\cdots $$ Since this value is composite when $n$ has at least two distinct prime factors, and 5 is prime, the only way to satisfy the right side being 5 is that $n$ has a single prime factor $p_1$ and $e_1=4$, i.e. $n = p^4$.

So what we need to do now is

write a program that goes through all primes after $^4\sqrt {20210318223247}$ and find the first one whose fourth power is a valid timestamp (has valid month, day, hour, minute, second).

Here is the code in Factor:

20210318223247 0.25 ^ 1 /i [ dup 4 ^
  { [ 10 8 ^ / floor 100 mod 1 12 between? ]
    [ 10 6 ^ / floor 100 mod 1 31 between? ]
    [ 10000 / floor 100 mod 24 < ]
    [ 100 / floor 100 mod 60 < ]
    [ 100 mod 60 < ] } 1&&
] [ next-prime ] do until dup . 4 ^ .

Try it online!

This code does not very rigorously check the validity of a timestamp, in that something like Feb 31 is not filtered out. Luckily, the first answer found has a well-formed month-day combination.

The output is

$2939^4 = 74610224073841$, which maps to 7461-02-24 07:38:41Z. It's way too distant in the future for us to wait for that time, as hexomino already pointed out :)

Answer 3

You need to post your answer on:

A time that is the fourth power of a prime number. The other answers have already posted the first one that works.

The way to find this is:

First, notice that it is somewhat difficult to have a number $n$ with an odd number of divisors. If $j$ is a divisor of $n$, then $\frac{n}{j} = k$ is also a divisor. The only way for divisors to not come in pairs is if you have a square number so that $j=k$. Thus our desired $n$ must be equal to the square of some number $m$. $m$ cannot be prime, otherwise $n$ would only have 3 factors. So $m$ has a factor $j$, meaning that $j$ and $mj$ are also factors of $n$. Thus the factors of $n$ are 1, $j$, $m$, $mj$, and $n$, so $j$ must be a prime number and $n=j^4$.

To make it even harder:

Most primes still won't work. For example, $2129^4=20544834434881$, but unless there are some significant changes in the next 30 years there will not be a 34th day of the 48th month in 2054.