# Erdős-Woods number

An integer $k$ is an Erdős-Woods number if there is an integer $n$ such that each of the consecutive integers $n+i$ for $0 shares at least one prime factor  with either $n$ or $n+k$. In other words, if for a $k$ there is an $n$ such that each evaluation of $\gcd(n,n+i)>1$ or $\gcd(n+k,n+i)>1$ returns true, then $k$ is an Erdős-Woods number.

For example, one $n$ for $k=16$ is 2184. 2184 is $2^{3}\times 3\times 7\times 13$, while $2184+16=2200=2^{3}\times 5^{2}\times 11$. We then verify that

• 2185 is clearly divisible by 5 and thus shares that odd prime as a factor with 2200.

• 2186 is even and so shares 2 as a factor with both 2184 and 2200.

• 2187 is 3 more than 2184 and therefore must also be divisible by 3. In fact, it is $3^{7}$.

• 2189 is 11 less than 2200 and therefore must be divisible by 11. In base 10 we can quickly verify that 2 + 8 = 10 and 1 + 9 is also 10.

• 2191 is 7 more than 2184 and thus must be divisible by 7.

• 2193 is 9 more than 2184 and thus divisible by 3.

• 2195 is obviously divisible by 5.

• 2197 is 13 more than 2184 and thus must be divisible by 13. In fact, it is $13^{3}$.

• 2199 is 15 more than 2184 and thus divisible by 3.

Knowing one $n$ for a given $k$ one can find other $n$ by multiplying the odd prime factors of $n+k$ (just once each, let’s call that product “$q$”) and then calculating $n(jq+1)$ with $j$ any positive integer of one’s choice. To give one example: with $n=2184$ and $j=17$, we get another $n$, namely 2044224. The range 2044224 to 2044240 displays the same patterns of factorization as described above, except that 2044227 and 2044237 are a semiprime and a sphenic number  respectively, as opposed to 2187 and 2197 which are both prime powers.

Other Erdős-Woods numbers are 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, etc., listed in A059756 of Sloane’s OEIS (the smallest odd Erdős-Woods number is 903), while A059757 lists the smallest matching $n$ for each of those $k$.

## References

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Title Erdős-Woods number ErdHosWoodsNumber 2013-03-22 17:37:14 2013-03-22 17:37:14 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A05 Erdos-Woods number Erdös-Woods number