I have to solve a couple of problems involving very large numbers, ie. those that are too big to use in a calculator. I clearly need a general approach for this sort of problem. Any pointers would be appreciated, ie. no need to solve the problems completely, just give me some direction. (If I get desperate I'll ask for actual solutions :)

The two problems are:

1) Is $2222^{5555} + 5555^{2222}$ prime?


2) Can you demonstrate that the digital representation of $7^{35}$ contains at least one digit that occurs four times?

Based on a hint from Chas, I have an idea for (2).

For a number to contain one digit four times, it must have at least 31 digits. That is, a 30-digit number may not meet the criterion as it could have each of the ten digits 3 times. But as soon as we have 31-digits, a fourth occurrence of one of the digits must be present.

Therefore we meet the criterion if:

\begin{align} 7^{35} &> 10^{30} \\ log(7^{35}) &> log(10^{30}) \\ 35 *log(7) &> 30 * log(10) \\ 35 * 0.845 &> 30 \\ 29.58 &> 30 \end{align}

The inequality is not true. Therefore I cannot prove the criterion can be met. Am I on the right track?


For Question 1, you can easily prove it is not prime by considering that both 5555 and 2222 are divisible by 1111

Continuing your proof on question 2:

You have shown correctly that $7^{35}$ has exactly 30 digits and that it is not enough to say there is at least one digit that repeats 4 times. By pigeon hole, the only time when this happens is when all the 10 digits repeat 3 times. So in this case, when we take the sum of the digits, it is a multiple of 3, and hence the number is divisible by 3.This is not possible has $7^{35}$ is not a multiple of 3. Thereby, we have proved that this case is not possible


Hint for the first question:

What is $2222^{5555} \mod 3$? What is $5555^{2222} \mod 3$?

Hint for the second question:

How many digits long is $7^{35}$?

  • $\begingroup$ Chas, thanks for the tips. You've given me an idea for the second Q which I've added to my question. Does it look right? Still no idea about the first one. $\endgroup$ – dave Oct 10 '16 at 10:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.