

A136585


Solutions of an a*x+b*y=c Prime Diophantine Equation: Prime[m]x+Prime[m+1]*y=Prime[m1] : as Abs[n*Prime[m]] Or Abs[n*Prime[m+1]] in x+y*n=Prime[m1].


0



2, 4, 5, 6, 9, 20, 33, 35, 42, 44, 57, 68, 104, 114, 117, 119, 145, 174, 279, 301, 310, 322, 345, 376, 410, 430, 517, 533, 590, 649, 740, 777, 976, 1159, 1537, 1590, 2345, 2412
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OFFSET

1,1


COMMENTS

Starting at the second prime 3, solutions are obtained to the Equation
x+y*n=Prime[m1]
or
n=(Prime[m1]+x)/y
Either n*Prime[m]/or n*Prime[m+1] is an Integer.
using the Wagon Diophantine solver Module for n and then the specific prime that it is a rational number of is multiplied out to give an Integer.
The resulting numbers are made positive and sorted for magnitude
to give the output sequence.
This sequence is an effort to get some sequence related by the primes:
{Prime[m1],Prime[m],Prime[m+1]}
by
Prime[m]x+Prime[m+1]*y=Prime[m1]


REFERENCES

A Course in Computational Number Theory by Bressoud and Wagon,2001


LINKS



FORMULA

a[out]=Abs[If[ IntegerQ[n*Prime[m+1]],n*Prime[m+1] else n*Prime[m]]] where n is a rational number: n=(Prime[m1]+x)/y Sequence is sorted by magnitude.


MATHEMATICA

Clear[n, m, l] DiophantineSolve[{a_, b_}, c_, n_] := Module[{d, e}, {d, e} = ExtendedGCD[a, b]; If[Mod[c, d] == 0, Transpose[{c*e, {b, a}}/d].{1, n}, {}]]; a = Table[Table[Simplify[If[l == 2, Prime[m], Prime[m + 1]]*(n /. Solve[DiophantineSolve[{Prime[m], Prime[m + 1]}, Prime[m  1], n][[l]]  Prime[m  1] == 0, n])], {l, 2, 1, 1}], {m, 2, 20}]; Union[Abs[Flatten[a]]]


CROSSREFS



KEYWORD

nonn,uned


AUTHOR



STATUS

approved



