Rough number

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

A k-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to k. k-roughness has alternately been defined as requiring all prime factors to strictly exceed k.[1]

Examples (after Finch)[edit]

  1. Every odd positive integer is 3-rough.
  2. Every positive integer that is congruent to 1 or 5 mod 6 is 5-rough.
  3. Every positive integer is 2-rough, since all its prime factors, being prime numbers, exceed 1.

See also[edit]


  1. ^ p. 130, Naccache and Shparlinski 2009.


  • Weisstein, Eric W. "Rough Number". MathWorld.
  • Finch's definition from Number Theory Archives
  • "Divisibility, Smoothness and Cryptographic Applications", D. Naccache and I. E. Shparlinski, pp. 115-173 in Algebraic Aspects of Digital Communications, eds. Tanush Shaska and Engjell Hasimaj, IOS Press, 2009, ISBN 9781607500193.

The On-Line Encyclopedia of Integer Sequences (OEIS) lists p-rough numbers for small p: