A306746 A Goldbug number is an even number 2m for which there exists a subset of the prime non-divisors, P={p1, p2, p3, ..., pk}, of 2m where (2m-p1)*(2m-p2)*(2m-p3)*...*(2m-pk) has only elements of P as factors and one of the pi is between m/2 and m for even m and between (m+1)/2 and m-1 for odd m.

128, 1718, 1862, 1928, 6142

Offset: 1

Author

Craig J. Beisel, Mar 07 2019

Keywords

• nonn
• more
• hard

Comments

A Goldbug number is an even number 2m for which there exists some subset of the prime non-divisors (PNDs) of 2m, 2 < p1 < p2 < p3 < ... < pk < m, such that (2m-p1)*(2m-p2)*(2m-p3)*...*(2m-pk) has only p1,p2,p3,...,pk as factors and one of the pi is between n/2 and n for even n and between (n+1)/2 and n-1 for odd n. We do not need to consider the case where n is prime, since then n itself is a Goldbach pair. A Goldbug number is called order k if the maximal subset satisfying the property is of size k. These numbers arise from Goldbug's Algorithm which attempts to find a Goldbach pair for a particular even number by starting with a given PND p1 and successively adding the factors of the product (2m - p1)*...*(2m - pk) to the search until a pair is found. Goldbug numbers are those even numbers for which Goldbug's Algorithm is not guaranteed to find a Goldbach pair since it could reach a subset of the PNDs which does not contain new information about additional PNDs to add to the search.

Goldbug numbers are a special case of Basic Pipes as defined by Wu. It has been shown computationally a(7) > 5*10^8. See link.

Goldbug numbers serve as a link between Goldbach's conjecture and the Pillai conjecture since order 2 Goldbug numbers represent solutions to its generalized difference equation. For example, sequence A057896 demonstrates there are no order 2 Goldbugs less than 10^24 since it would imply additional solutions to the equation a^x-a = b^y-b. In fact, theorem 3 from Scott[1993] implies that no additional order 2 Goldbugs exist at all.

Examples

			Although 2200 and the prime non-divisors 3 and 13 might seem to satisfy the definition since (2200 - 13)*(2200 - 3) = 4804839 = 3^7*13^3, 2200 is not an order k=2 Goldbug since neither 3 or 13 is in the interval (n/2,n).
A higher-order example is the term 128, for which there exists a subset of the PNDs such that the corresponding product (128 - 3)*(128 - 5)*(128 - 7)*(128 - 11)*(128 - 13)*(128 - 17)*(128 - 23)*(128 - 29)*(128 - 37)*(128 - 41)*(128 - 43)*(128 - 47)*(128 - 53)*(128 - 59) = 8147166895749452778629296875 = (3^14)*(5^8)*(7^2)*(11^3)*(13^2)*17*(23^2)*29*37*41 and 37 and 41 are in the interval (32,64). Therefore, 128 is a Goldbug number of order k=14.
		

Links

• Craig J. Beisel, Maximal Sets of PNDs Satisfying Goldbug Property for First 5 Terms
• Craig J. Beisel, Enumeration of all Goldbug subsets for the term 128.
• Craig J. Beisel, Goldbug's Algorithm.
• Andrzej Bożek, Exceptional autonomous components of Goldbach factorization graphs, arXiv:1909.09900 [math.NT], 2019.
• Bert Dobbelaere, C++ Program.
• Christian Goldbach, Letter to L. Euler, June 7, 1742.
• Mathematics Stack Exchange, Searching for Goldbug Numbers.
• Reese Scott, On the Equations p^x - b^y = c and a^x + b^y = c^z, Journal of Number Theory, Volume 44, Issue 2, June 1993.
• Willie Wu, Pipe Theory.
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A244408, A057896.

Programs

PARI

isgbk(n,k) = {if (n % 2, return (0)); f=factor(n) [, 1]; vp = setminus(primes([3, n/2]), f~); forsubset([#vp,k], s, w=vecextract(vp, s); if(#w>1 && setminus(factor(x=prod(i=1, #s, n-w[i]))[, 1]~, Set(w))==[], return(1)););return(0);} \\ tests if n is order k Goldbug;