# 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
Number Theory
Medium
##### Suggested Book
An Excursion in Mathematics

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 $\{(1,1),(2,1),(4,2)\}$.

# 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

## Isomorphism in b/w infinite dim vector sp: TIFR GS 2019, Part B Problem 10

It is a question on isomomorphisms b/w inf dim vector spaces. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Matrix to real line: TIFR GS 2019, Part B Problem 8

It is a lie algebra question on connections b/w matrices and real space. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Homomorphism to Continuous function: TIFR GS 2019, Part B Problem 9

It is a lie algebra question on homomorphisms b/w real ring and ring of continuous function. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Similar matrices: TIFR GS 2019, Part B Problem 7

It is a linear algebra question on similar matrices. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Average Determinant: TIFR GS 2017 Part A Problem 8.

This question has appeared in TIFR GS 2017 Entrance Examination and is based on Linear Algebra.

## Spanning set of a matrix space: TIFR GS 2019, Part B Problem 6

It is a linear algebra question on matrices basically on spanning set of a matrix space. It was asked in TIFR 2019 GS admission paper.

## ABC of rank: TIFR GS 2019, Part B Problem 5

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Continuous map on countable space: TIFR GS 2019, Part B Problem 3

It is a topology question on real plane. It was asked in TIFR 2019 GS admission paper. It is true-false type question.

## Invertible Matrix implies identity?: TIFR GS 2019, Part B Problem 4

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper.

## Invertible Matrix: TIFR GS 2019, Part B Problem 2

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper.