# Understand the problem

Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

##### Source of the problem

Bangladesh MO 2019 Problem 1

##### Topic

Number Theory

##### Difficulty Level

5/10

##### Suggested Book

A Friendly Introduction to Number Theory by J.H.Silverman

# Start with hints

Do you really need a hint? Try it first!

Write the problem in a Mathematical Equation form i.e. . Now can you like factorize the stuff to make life easier and use divisibility rules?

After factorizing, we get . Now can use the prime factorization idea and see what are the cases possible. Observe that three cases are possible:

Observe that a, b are both positive integers. Hence the case: is absurd. Let’s concentrate on the other cases one by one. Now,observe this that , which is has a solution iff a = b = 1. What about the other case?

Observe a = – b (mod p ) this together with the second equation gives

(modp). Now p can be 3. For p = 3, Observe that a = 1 and b = 2 is a solution. Now if p is not 3, then p must divide a and b. This implies a + b must be greater than equal to 2p, hence contradiction.

(modp). Now p can be 3. For p = 3, Observe that a = 1 and b = 2 is a solution. Now if p is not 3, then p must divide a and b. This implies a + b must be greater than equal to 2p, hence contradiction.

Hence the solutions are a = 1, b =1, p = 2

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