Select Page

# 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.
Number Theory
5/10
##### Suggested Book
A Friendly Introduction to Number Theory by J.H.Silverman

Do you really need a hint? Try it first!

Write the problem in a Mathematical Equation form i.e. $p^2 = a^3 + b^3$. Now can you like factorize the stuff to make life easier and use divisibility rules?
After factorizing, we get  $p^2 = (a+b)(a^2 + b^2 - ab)$. Now can use the prime factorization idea and see what are the cases possible. Observe that three cases are possible:
• $a+b = p, a^2 + b^2 - ab =p$
• $a+b =p^2 , a^2 + b^2 - ab = 1$
• $a+b = 1, a^2 + b^2 - ab = p^2$
Now, can you decode these cases and solve the problem like Sherlock?
Observe that a, b are both positive integers. Hence the case: $a+b = 1, a^2 + b^2 - ab = p^2$ is absurd. Let’s concentrate on the other cases one by one. $a+b =p^2 , a^2 + b^2 - ab = 1$ Now,observe this that $a^2 + b^2 - ab = (a-b)^2 + ab = 1$, which is has a solution iff a = b = 1. What about the other case? $a+b = p , a^2 + b^2 - ab = p$
$a+b = p, a^2 + b^2 - ab =p$ Observe a = – b (mod p ) this together with the second equation gives
$3a^2 = 0$ (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

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Tetrahedron Problem | AIME I, 1992 | Question 6

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Tetrahedron Problem.

## Triangle and integers | AIME I, 1995 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Triangle and integers.

## Functional Equation Problem from SMO, 2018 – Question 35

Try this problem from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation. You may use sequential hints if required.

## Sequence and greatest integer | AIME I, 2000 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Sequence and the greatest integer.

## Arithmetic sequence | AMC 10A, 2015 | Problem 7

Try this beautiful problem from Algebra: Arithmetic sequence from AMC 10A, 2015, Problem. You may use sequential hints to solve the problem.

## Series and sum | AIME I, 1999 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Series and sum.

## Inscribed circle and perimeter | AIME I, 1999 | Question 12

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.

## Problem based on Cylinder | AMC 10A, 2015 | Question 9

Try this beautiful problem from Mensuration: Problem based on Cylinder from AMC 10A, 2015. You may use sequential hints to solve the problem.

## Cubic Equation | AMC-10A, 2010 | Problem 21

Try this beautiful problem from Algebra, based on the Cubic Equation problem from AMC-10A, 2010. You may use sequential hints to solve the problem.

## Median of numbers | AMC-10A, 2020 | Problem 11

Try this beautiful problem from Geometry based on Median of numbers from AMC 10A, 2020. You may use sequential hints to solve the problem.