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number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...
, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s one can assign a distinct prime that divides it. It was first published in ''
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
'', 76(1969) 1126-1128.


Formal statement

If ''n'' + 1, ''n'' + 2, …, ''n'' + ''k'' are all
composite numbers A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
, then there are ''k'' distinct primes ''p''''i'' such that ''p''''i''
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
''n'' + ''i'' for 1 ≤ ''i'' ≤ ''k''.


Weaker version

A weaker, though still unproven, version of this conjecture states: If there is no prime in the interval +1, n+k/math>, then \prod_(n+x) has at least ''k'' distinct prime divisors.


See also

* Prime gap


References

* * * Guy, R. K. "Grimm's Conjecture." §B32 in ''Unsolved Problems in Number Theory'', 3rd ed.,
Springer Science+Business Media Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, pp. 133–134, 2004. * * * * * * * *


External links


Prime Puzzles #430
{{Prime number conjectures Conjectures about prime numbers Unsolved problems in number theory