Grimm's conjecture

Apr 20, 2026 - 13:59
0 0
Grimm's conjecture

doi-access=free for a reference + M. Ram Murty + footnotes

← Previous revision Revision as of 08:28, 20 April 2026
Line 1: Line 1:
{{Short description|Prime number conjecture}}
{{Short description|Prime number conjecture}}
In [[mathematics]], specifically in [[number theory]], '''Grimm's conjecture''' states that, for every set of consecutive [[composite number]]s, there is an equally sized set of [[prime number]]s, and a [[bijection]] that maps each composite in the former set to a prime in the latter set that it is divisible by. It was first proposed by Carl Albert Grimm in 1969.{{Cite journal |last=Grimm |first=C. A. |date=1969 |title=A conjecture on consecutive composite numbers |journal=American Mathematical Monthly |volume=76 |issue=10 |pages=1126–1128 |doi=10.1142/S1793042106000498 |jstor=2317188}}
In [[mathematics]], specifically in [[number theory]], '''Grimm's conjecture''' states that, for every set of consecutive [[composite number]]s, there is an equally sized set of [[prime number]]s, and a [[bijection]] that maps each composite in the former set to a prime in the latter set that it is divisible by. It was first proposed by Carl Albert Grimm in 1969.{{sfn|Grimm|1969}}


Though still unproven, the conjecture has been verified for all n<1.9\times 10^{10}.{{Cite journal |last=Laishram |first=Shanta |last2=Shorey |first2=T. N. |date=2006 |title=Grimm's conjecture on consecutive integers |journal=International Journal of Number Theory |volume=2 |issue=2 |pages=207–211 |doi=10.1142/S1793042106000498}}
Though still unproven, the conjecture has been verified for all n<1.9\times 10^{10}.{{sfn|Laisham|Shorey|2006}}


== Formal statement ==
== Formal statement ==
Line 12: Line 12:
A weaker, though still unproven, version of this conjecture states that if there is no prime in the interval [n+1, n+k], then
A weaker, though still unproven, version of this conjecture states that if there is no prime in the interval [n+1, n+k], then
:\prod_{1\,\leq \,x\,\leq \,k}(n+x)
:\prod_{1\,\leq \,x\,\leq \,k}(n+x)
has at least k distinct [[prime divisor]]s.{{Cite journal |last=Erdős |first=P. |last2=Selfridge |first2=J. L. |date=1971 |title=Some problems on the prime factors of consecutive integers II |journal=Proceedings of the Washington State University Conference on Number Theory |pages=13–21}}
has at least k distinct [[prime divisor]]s.{{sfn|Erdős|Selfridge|1971}}


== Consequences ==
== Consequences ==
Line 18: Line 18:
If Grimm's conjecture is true, then
If Grimm's conjecture is true, then
:p_{i+1} - p_i \ll \Big(\frac{p_i}{\log p_i}\Big)^{1/2}
:p_{i+1} - p_i \ll \Big(\frac{p_i}{\log p_i}\Big)^{1/2}
for all consecutive primes p_i and p_{i+1}. This goes well beyond what the [[Riemann hypothesis]] would imply about [[prime gaps|gaps between prime numbers]]: the Riemann hypothesis only implies an upper bound of O(p_i^{1/2}(\log p_i)).{{Cite journal |last=Laishram |first=Shanta |last2=Murty |first2=M. Ram |date=2012 |title=Grimm's conjecture and smooth numbers |url=https://projecteuclid.org/journals/michigan-mathematical-journal/volume-61/issue-1/Grimms-conjecture-and-smooth-numbers/10.1307/mmj/1331222852.full |journal=Michigan Mathematical Journal |volume=61 |issue=1 |pages=151–160 |doi=10.1307/mmj/1331222852 |arxiv=1306.0765}}
for all consecutive primes p_i and p_{i+1}.{{sfn|Erdős|Selfridge|1971}} This goes well beyond what the [[Riemann hypothesis]] would imply about [[prime gaps|gaps between prime numbers]]: the Riemann hypothesis only implies an upper bound of O(p_i^{1/2}(\log p_i)).{{sfn|Laishram|Murty|2012}}


== See also ==
== See also ==
Line 27: Line 27:


== References ==
== 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}}
*{{cite journal |last1=Erdős |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}}


*{{cite journal |last1=Grimm |first1=C. A. |title=A conjecture on consecutive composite numbers |journal=American Mathematical Monthly |date=1969 |volume=76 |issue=10 |pages=1126–1128 |doi=10.2307/2317188 |jstor=2317188}}
*{{cite journal |last1=Grimm |first1=C. A. |title=A conjecture on consecutive composite numbers |journal=American Mathematical Monthly |date=1969 |volume=76 |issue=10 |pages=1126–1128 |doi=10.2307/2317188 |jstor=2317188}}


*[[Richard K. Guy|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}}
*[[Richard K. Guy|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 |last=Laishram |first=Shanta |last2=Murty |first2=M. Ram |authorlink2=M. Ram Murty |date=2012 |title=Grimm's conjecture and smooth numbers |journal=[[Michigan Mathematical Journal]] |volume=61 |issue=1 |pages=151–160 |doi=10.1307/mmj/1331222852 |doi-access=free |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=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}}
Read More