Understand the problem

Find all pairs of positive integers $ (n,k)$ so that $ (n+1)^k-1=n!$.

Source of the problem
Singapore MO 2008
Topic
Number Theory
Difficulty Level
Medium
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

Note that n+1 has to be a prime, because any proper prime divisor of n+1 would divide n! too.
Say n+1=p. Note that, for large enough p, both \frac{p-1}{2} and 2 are factors of (p-1)!. Thus, the factor (p-1) occurs twice in the expansion of (p-1)!.
Prove that, if (p-1)^2|p^k-1 then p-1|k.
Hint 3 implies that p-1\le k. Hence p^{p-1}-1\le p^k-1=(p-1)!. This is obviously false. Hence, p has to be small enough to avoid this situation. It is avoided precisely when p\in\{2,3,5\}. This corresponds to n\in\{1,2,4\}. The equations to be solved are 2^k=2, 3^k=3 and 5^k=25. Hence, the solutions are \{(1,1),(2,1),(4,2)\}.

Watch the video (Coming Soon)

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 2a

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

Data, Determinant and Simplex

This problem is a beautiful problem connecting linear algebra, geometry and data. Go ahead and dwelve into the glorious connection.

Problem on Integral Inequality | ISI – MSQMS – B, 2015

Try this problem from ISI MSQMS 2015 which involves the concept of Integral Inequality and real analysis. You can use the sequential hints provided to solve the problem.

Inequality Problem From ISI – MSQMS – B, 2017 | Problem 3a

Try this problem from ISI MSQMS 2017 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

Problem on Natural Numbers | TIFR B 2010 | Problem 4

Try this problem of TIFR GS-2010 using your concepts of number theory and congruence based on natural numbers. You may use the sequential hints provided.

Definite Integral Problem | ISI 2018 | MSQMS- A | Problem 22

Try this problem from ISI-MSQMS 2018 which involves the concept of Real numbers, sequence and series and Definite integral. You can use the sequential hints

Inequality Problem | ISI – MSQMS 2018 | Part B | Problem 4

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality and Combinatorics. You can use the sequential hints provided.

Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 4b

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

Positive Integers Problem | TIFR B 201O | Problem 12

Try this problem of TIFR GS-2010 using your concepts of number theory based on Positive Integers. You may use the sequential hints provided.

CYCLIC GROUP Problem | TIFR 201O | PART A | PROBLEM 1

Try this problem from TIFR GS-2010 which involves the concept of cyclic group. You can use the sequential hints provided to solve the problem.