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Grimm's conjecture

{{Short description|Prime number conjecture}}

In number theory, 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 numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.

Formal statement

If n + 1, n + 2, ..., n + k are all composite numbers, then there are k distinct primes pi such that pi divides 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 [n+1, n+k], then \prod_{1\le x\le k}(n+x) has at least k distinct prime divisors.

See also

References

  • {{cite journal|last1=Erdös|first1=P.|last2=Selfridge|first2=J. L.|title=Some problems on the prime factors of consecutive integers II|journal=Proceedings of the Washington State University Conference on Number Theory|date=1971|pages=13–21|url=https://old.renyi.hu/~p_erdos/1971-24.pdf}}
  • {{cite journal|last1=Grimm|first1=C. A.|title=A conjecture on consecutive composite numbers|journal=The American Mathematical Monthly|date=1969|volume=76|issue=10|pages=1126–1128|doi=10.2307/2317188|jstor=2317188}}
  • Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 3rd ed., Springer Science+Business Media, pp. 133–134, 2004. {{ISBN|0-387-20860-7}}
  • {{cite journal|last1=Laishram|first1=Shanta|last2=Murty|first2=M. Ram|title=Grimm's conjecture and smooth numbers|journal=The Michigan Mathematical Journal|date=2012|volume=61|issue=1|pages=151–160|doi=10.1307/mmj/1331222852|url=https://projecteuclid.org/euclid.mmj/1331222852|arxiv=1306.0765}}
  • {{cite journal|last1=Laishram|first1=Shanta|last2=Shorey|first2=T. N.|title=Grimm's conjecture on consecutive integers|journal=International Journal of Number Theory|date=2006|volume=2|issue=2|pages=207–211|doi=10.1142/S1793042106000498|url=https://www.worldscientific.com/doi/abs/10.1142/S1793042106000498|url-access=subscription}}
  • {{cite journal|last1=Ramachandra|first1=K. T.|last2=Shorey|first2=T. N.|last3=Tijdeman|first3=R.|title=On Grimm's problem relating to factorisation of a block of consecutive integers|journal=Journal für die reine und angewandte Mathematik|date=1975|volume=273|pages=109–124|doi=10.1515/crll.1975.273.109}}
  • {{cite journal|last1=Ramachandra|first1=K. T.|last2=Shorey|first2=T. N.|last3=Tijdeman|first3=R.|title=On Grimm's problem relating to factorisation of a block of consecutive integers. II|journal=Journal für die reine und angewandte Mathematik|date=1976|volume=288|pages=192–201|doi=10.1515/crll.1976.288.192}}
  • {{cite journal|last1=Sukthankar|first1=Neela S.|title=On Grimm's conjecture in algebraic number fields|journal=Indagationes Mathematicae (Proceedings)|date=1973|volume=76|issue=5|pages=475–484|doi=10.1016/1385-7258(73)90073-5|doi-access=}}
  • {{cite journal|last1=Sukthankar|first1=Neela S.|title=On Grimm's conjecture in algebraic number fields. II|journal=Indagationes Mathematicae (Proceedings)|date=1975|volume=78|issue=1|pages=13–25|doi=10.1016/1385-7258(75)90009-8|doi-access=}}
  • {{cite journal|last1=Sukthankar|first1=Neela S.|title=On Grimm's conjecture in algebraic number fields-III|journal=Indagationes Mathematicae (Proceedings)|date=1977|volume=80|issue=4|pages=342–348|doi=10.1016/1385-7258(77)90030-0|doi-access=}}
  • {{mathworld|urlname=GrimmsConjecture|title=Grimm's Conjecture}}

External links

  • [http://www.primepuzzles.net/puzzles/puzz_430.htm Prime Puzzles #430]

{{Prime number conjectures}}

Category:Conjectures about prime numbers

Category:Unsolved problems in number theory