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. 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 

Watch video

Connected Program at Cheenta

Math Olympiad Program

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

Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.

Rectangular Piece of Paper | AMC 10A, 2014| Problem No 22

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2014. Problem-23. You may use sequential hints to solve the problem.

Probability in Marbles | AMC 10A, 2010| Problem No 23

Try this beautiful Problem on Probability from AMC 10A, 2010. Problem-23. You may use sequential hints to solve the problem.

Points on a circle | AMC 10A, 2010| Problem No 22

Try this beautiful Problem on Number theory based on Triangle and Circle from AMC 10A, 2010. Problem-22. You may use sequential hints to solve the problem.

Circle and Equilateral Triangle | AMC 10A, 2017| Problem No 22

Try this beautiful Problem on Triangle and Circle from AMC 10A, 2017. Problem-22. You may use sequential hints to solve the problem.