Brocard–Ramanujan problem

Multiply $1\times2\times3\times4$ to get 24, then add one: $24+1=25=5^{2}$. Multiplying all positive integers through n gives its factorial, written $n!$. Brocard–Ramanujan asks when this product plus one is a perfect square, the product of a whole number with itself.

The known successes are n equal to 4, 5 and 7, giving squares of 5, 11 and 71. Are there any other positive integer inputs? The question covers all of them, however large.[1]

When can the product of all integers from 1 through n be exactly one less than an integer square? The Brocard–Ramanujan problem asks for every positive integer solution of $n!+1=m^{2}$, without restricting m to a special sequence or changing the number system. The known pairs are $(4,5)$, $(5,11)$ and $(7,71)$.

Subtracting one from the square gives a useful factorization: $$n!=(m-1)(m+1).$$ For n at least 2, the factorial is even, so m must be odd. The two displayed factors are then even, differ by 2, and have 2 as the largest positive integer dividing both. For the first known solution, they are 4 and 6, whose product is 24. Thus their odd prime factors cannot be shared, and the factorial's prescribed prime powers must fit this closely spaced pair.

These are elementary necessary conditions derived from the equation. They do not exclude every larger n. The selected sequence paper uses more specific information about prime-power divisibility when m lies in a Padovan or Perrin recurrence sequence; that added structure supplies restrictions unavailable for arbitrary m.[1]

No proof excludes every further positive integer solution.[1]

The June preprint’s reported Padovan/Perrin result restricts m. The published finite-field paper changes the ambient algebra and factorial definition.[1] [2]

Factorials collect every small factor into one product, while squares use the same integer twice. Being exactly one apart brings these very different patterns together. The problem tests how adding a single unit can change a number's multiplicative structure.[1]

Near a square, the factorial's many small prime factors must split between two neighboring even integers. A way to control that split for unrestricted m would address the interaction of additive spacing and multiplicative divisibility, a recurring issue in integer equations.

The restricted sequence results illustrate one route: compare how many copies of a prime divide the factorial with how many divide the neighboring sequence values. The finite-field analogue explores related questions in a different arithmetic setting. Those connections motivate the integer problem without turning a solved variant or a finite search into a completeness proof.[1] [2]