OFFSET
1,10
COMMENTS
This is inspired by the sequence A154290 and related conjectures of Sun. On Jan 08 2009, Zhi-Wei Sun and Qing-Hu Hou conjectured that a(n)>0 for n=6,7,...; in other words, any integer n>5 can be written as the sum of an odd prime, a positive Pell number and a companian Pell number. The Pell numbers are defined by P_0=0, P_1=1 and P_{n+1}=2P_n+P_{n-1} (n=1,2,3,...) and the companion Pell numbers are given by Q_0=Q_1=2 and Q_{n+1}=2Q_n+Q_{n-1} (n=1,2,3...). Note that for n>5 both P_n and Q_n are greater than 2^n.
D. S. McNeil disproved the conjecture by finding the 4 initial counterexamples: 169421772576, 189661491306, 257744272674, 534268276332. - Zhi-Wei Sun, Jan 17 2009
On Feb 01 2009, Zhi-Wei Sun observed that these 4 counterexamples are divisible by 42 and guessed that all counterexamples to the conjecture of Sun and Hou should be multiples of 42. - Zhi-Wei Sun, Feb 01 2009
LINKS
Zhi-Wei Sun, Table of n, a(n), n=1..100000.
R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
D. S. McNeil, Sun's strong conjecture, NMBRTHRY, Dec 2008.
Z. W. Sun, A congruence for primes, Proc. Amer. Math. Soc. 123(1995), 1341-1346.
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t, NMBRTHRY, Dec 2008.
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t, NMBRTHRY, Dec 2008.
Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010.
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
EXAMPLE
For n=10 the a(10)=3 solutions are 3+5+2, 3+1+6, 7+1+2.
MAPLE
Pell:=proc(n) if n=0 then return(0); elif n=1 then return(1); else return( 2*Pell(n-1) + Pell(n-2) ); fi; end proc: comp_Pell:=proc(n) if n=0 then return(2); elif n=1 then return(2); else return( 2*comp_Pell(n-1) + comp_Pell(n-2) ); fi; end proc: for n from 1 to 10^5 do rep_num:=0; for i from 1 while Pell(i)<n do for j from 1 while Pell(i)+comp_Pell(j)<n do p:=n-Pell(i)-comp_Pell(j); if (p>2) and isprime(p) then rep_num:=rep_num+1; fi; od; od; printf("%d %d\n", n, rep_num); od:
MATHEMATICA
nmax = 10^3;
Pell[n_] := Pell[n] = If[n == 0, Return[0], If[n == 1, Return[1], Return[2* Pell[n - 1] + Pell[n - 2]]]];
compPell[n_] := compPell[n] = If[n == 0, Return[2], If[n == 1, Return[2], Return[2*compPell[n - 1] + compPell[n - 2]]]];
Reap[For[n = 1, n <= nmax, n++, repnum = 0; For[i = 1, Pell[i] < n, i++, For[j = 1, Pell[i] + compPell[j] < n, j++, p = n - Pell[i] - compPell[j]; If[p > 2 && PrimeQ[p], repnum++]]]; Sow[repnum]]][[2, 1]] (* Jean-François Alcover, Dec 13 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 07 2009
STATUS
approved