Try this beautiful problem from the Pre-RMO, 2017, Question 23, based on Solving Equation.
Suppose an integer r, a natural number n and a prime number p satisfy the equation \(7x^{2}-44x+12=p^{n}\). Find the largest value of p.
Equation
Algebra
Integers
Answer: is 47.
PRMO, 2017, Question 23
Higher Algebra by Hall and Knight
\(7x^{2}-44x+12=p^{n}\)
or, \(7x^{2}-42x-2x+12=p^{n}\)
or, \((7x-2)(x-6)=p^{n}\)
or, \(7x-2=p^{\alpha}\), \(x-6=p^{\beta}\)
or, \((7x-2)-7(x-6)\)=\(p^{\alpha}-7p^{\beta}\)
\(40=p^{\alpha}-7p^{\beta}\)
If \({\alpha}, {\beta}\) are natural numbers, p is a divisor of 40
or, p=2 or 5
If p=2, 40=\(2^{\alpha}-7(2^{\beta})\) or, \((2^{3})(5)\)=\(2^{\alpha}-7(2^{\beta})\)
or, \({\beta}\)=3 and \(2^{\alpha}\)=40+56
or, \({\alpha}\) not an integer hence not possible
If p=5 then 40=\(5^{\alpha}-7(5^{\beta})\)
or, \((2)^{3}(5)=5^{\alpha}-7(5^{\beta})\)
or, \({\beta}=1\) and \(5^{\alpha}\)=40+35
or, \({\alpha}\) not an integer hence not possible
so \({\beta}\)=0 or, \(p^{\alpha}\)=47
or, p=47 and \({\alpha}\)=1.
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]