Erdős-Woods conjecture

This article deals with number theoretic conjecture. For the thematically related definition see Erdős-Woods number .

The Erdős-Woods conjecture from number theory by Alan Robert Woods , established in his dissertation in 1981 , states:

Any integer is given . Then there is a positive integer such that it is uniquely determined by the list of prime factors of . ${\displaystyle n\geq 2}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle n+1,\dotsc ,n+k}$

Note that only the list of prime factors is given, not their multiplicity. The presumption is only proven in special cases.

If one denotes the set of the different prime factors of , it can also be formulated as follows: ${\displaystyle rad(m)}$ ${\displaystyle m}$

There are a whole number , so that made for follows . ${\displaystyle k\geq 3}$ ${\displaystyle rad(n+i)=rad(m+i)}$ ${\displaystyle i=1,2,\dotsc ,k}$ ${\displaystyle m=n}$

The conjecture is also named after Paul Erdős , who made a conjecture in 1980 from which the Erdős-Woods conjecture followed (as Woods explicitly noted).

Examples

It applies that counterexamples can be given for smaller values. ${\displaystyle k\geq 3}$

• Counterexample for : ${\displaystyle k=1}$

Be and . ${\displaystyle n:=8=2^{3}}$ ${\displaystyle m:=26=2\cdot 13}$

Then is and and thus . ${\displaystyle n+1=9=3^{2}}$ ${\displaystyle m+1=27=3^{3}}$ ${\displaystyle rad(n+1)=rad(m+1)=\{3\}}$

It is , thus a counterexample of the Erdős-Woods conjecture was found for the case . ${\displaystyle m\not =n}$ ${\displaystyle k=1}$

• Counterexample for : ${\displaystyle k=2}$

Be and . ${\displaystyle n:=74=2\cdot 37}$ ${\displaystyle m:=1214=2\cdot 607}$

Then is and and thus . ${\displaystyle n+1=75=3\cdot 5^{2}}$ ${\displaystyle m+1=1215=3^{5}\cdot 5}$ ${\displaystyle rad(n+1)=rad(m+1)=\{3,5\}}$

Furthermore is and and thus . ${\displaystyle n+2=76=2^{2}\cdot 19}$ ${\displaystyle m+2=1216=2^{6}\cdot 19}$ ${\displaystyle rad(n+2)=rad(m+2)=\{2,19\}}$

It is , thus a counterexample of the Erdős-Woods conjecture was found for the case . ${\displaystyle m\not =n}$ ${\displaystyle k=2}$

• Example for : ${\displaystyle k=3}$

Be . ${\displaystyle n:=26=2\cdot 13}$

Then is and thus . ${\displaystyle n+1=27=3^{3}}$ ${\displaystyle rad(n+1)=\{3\}}$

It is and thus . ${\displaystyle n+2=28=2^{2}\cdot 7}$ ${\displaystyle rad(n+2)=\{2,7\}}$

Furthermore is a prime number and thus . ${\displaystyle n+3=29\in \mathbb {P} }$ ${\displaystyle rad(n+3)=\{29\}}$

There is no other natural number actually , so to apply. The reason for this is the following sentence : ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle rad(n+i)=rad(m+i)}$ ${\displaystyle i=1,2,3}$

Let be a prime number and . Then the Erdős-Woods conjecture is true with . ${\displaystyle p\geq 5}$ ${\displaystyle n=p-3}$ ${\displaystyle k=3}$

useful information

The conjecture follows from the unproven abc conjecture : Michel Langevin proved in 1993, assuming the abc conjecture, that for with a constant . The Erdős-Woods conjecture is fulfilled by an infinite number of whole numbers . Next is even for the number of positive integers that satisfy the conjecture with at least one constant (Subburam, Thangadurai). ${\displaystyle k=3}$ ${\displaystyle n>C}$ ${\displaystyle C}$ ${\displaystyle n}$ ${\displaystyle k=2}$ ${\displaystyle n\leq x}$ ${\displaystyle d{\frac {x}{\log x}}}$ ${\displaystyle d>2}$

Woods' dissertation was about whether multiplication can be defined in a logical language over natural numbers, in which there is the predicate of the successor of a natural number and the predicate that two numbers have no common prime factors (the problem came from Julia Robinson ). Woods proved that this is equivalent to the Erdős-Woods conjecture. He also proved that the definability of multiplication is equivalent to the definability of addition, equality and the less than or equal relation.

Individual evidence

1. ↑ Woods was Adjunct Professor at the University of Western Australia and died in 2011. Vale Alan Woods, Australian Mathematical Society
2. ↑ Woods, Some problems in logic and number theory, and their connections, Dissertation, University of Manchester 1981. Doctoral advisor was Jeff Paris ( https://genealogy.math.ndsu.nodak.edu/id.php?id=59312 )
3. ↑ Erdős, How many pairs of products of consecutive integers have the same prime factors ?, American Mathematical Monthly , Volume 87 (5), 1980, pp. 391-392
4. ↑ S. Subburam, R. Thangadurai: On Erdős-Wood's conjecture , Proc. Indian Acad. Sci. , Volume 125 , 2015, pp. 139-147, Corrollary 1.1 at p. 140
5. ↑ Michel Langevin, Cas d'égalite pour le théorème de Mason et applications de la conjecture (abc), CR Acad. Sci., Paris, Ser. I, Vol. 317 (5), 1993, pp. 441-444
6. ↑ For the proof see Jörn Steuding, Diophantine Analysis, Chapman and Hall 2005, p. 186
7. ↑ S. Subburam, R. Thangadurai: On Erdős-Wood's conjecture , Proc. Indian Acad. Sci. , Volume 125 , 2015, pp. 139-147
8. ↑ Woods, Dissertation, p. 53