# When should this question be answered?

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.

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.

• FYI: At least on my browser - Chrome on a desktop, the time appears in the top-left, not the bottom-right. Is it different for the mobile version? – Darrel Hoffman Mar 19 at 13:35
• @DarrelHoffman: It's in both places, you can see it above the username in the bottom-right of the question too. Hover the "asked 16 hours ago" label (of whatever it may say at time of reading this). – musefan Mar 19 at 15:31
• Whoa, what? You can view the timestamp? I've been on Stack Exchange for over 8 years and this is the first time I've heard of this feature, lol I feel dumb... – QBrute Mar 21 at 0:35

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).

• Good job! Don't forget to come back for the bounty when it's the right time ^_^ – WhatsUp Mar 18 at 23:34
• No one wants to wait around for 918 years just for 500 rep points. So I am writing a script to do it... – A. I. Breveleri Mar 20 at 3:56

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 :)

• Ah ... very well-written answer! However you are just a tiiiiny bit slower than hexomino. I hope you understand that I give the green check to hexomino. Of course you still get my upvote! – WhatsUp Mar 18 at 23:36

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.

• Notice that there is a computer-puzzle tag, which means that it is intended to be solved using a computer. – WhatsUp Mar 18 at 23:40
• @WhatsUp I was writing up my explanation of why the number need to have the property that it does when Hexomino answered. I decided not to bother finishing my program to find it because I was already beaten to the punch. – Rob Watts Mar 18 at 23:45
• Oh yes, I appreciate your detailed explanation (+1). – WhatsUp Mar 18 at 23:49