Grimm's conjecture
Encyclopedia
In mathematics
, and in particular number theory
, Grimm's conjecture states that to each element of a set of consecutive composite number
s one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly
, 76(1969) 1126-1128.
n + i for 1 ≤ i ≤ k.
, then 
has at least k distinct prime divisors.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Grimm's conjecture states that to each element of a set of consecutive composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
s one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
, 76(1969) 1126-1128.
Formal statement
Suppose n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes pi such that pi dividesDivisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
n + i for 1 ≤ i ≤ k.
Weaker version
A weaker, though still unproven, version of this conjecture goes: If there is no prime in the interval, then has at least k distinct prime divisors.