OFFSET
1,1
COMMENTS
It was conjectured by Recamán Santos in 1976 and proved by Hausman and Shapiro in 1978 that 12 is the largest k possible.
Pomerance & Penney (1977) reported in a letter that they have proved that the conjecture is true. - Amiram Eldar, May 15 2020
REFERENCES
Paulo Ribenboim, The New Book of Prime Number Records, Third ed., Springer-Verlag New York, 1996, p. 42.
LINKS
Miriam Hausman and Harold N. Shapiro, Adding totitives, Mathematics Magazine, Vol. 51, No. 5 (1978), pp. 284-288.
Carl Pomerance and David E. Penney, Santos' conjecture, News & Letters, Mathematics Magazine, Vol. 50, No. 2 (1977), p. 107.
Bernardo Recamán Santos, Twelve and its totatives, Mathematics Magazine, Vol. 49, No. 5 (1976), pp. 239-240.
EXAMPLE
For k = 12 the numbers j are {1,5,7,11} and the numbers k+j are {13,17,19,23}, which are all prime.
MATHEMATICA
sQ[n_/; n>1]:=AllTrue[n+Select[Range[n-1], GCD[#, n]==1&], PrimeQ]; Select[Range[12], sQ]
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Ivan N. Ianakiev, Jul 26 2019
STATUS
approved