10 Jul
2006
10 Jul
'06
1:53 p.m.
(I hope I am not embarrassing myself with this one...) In an analogy with Zeckendorf (Fibonacci) number systems, can every positive integer be expressed as a _sum_ of _distinct_ (positive) primes ? If not, what is the smallest integer not so expressible? For this purpose, I'll allow "1" as a "prime", although I suspect it's only needed a constant number of times. 2=2 3=3 4=3+1 5=5 6=5+1 7=7 8=5+3 9=7+2 10=7+3 11=11 12=7+5 13=13 14=11+3 15=13+2 16=13+3 17=17 18=13+5 19=19 20=17+3 21=19+2 22=19+3 23=23 24=19+5 25=23+2 26=23+3 27=19+5+3 28=23+5 29=29 etc. I'm not (yet) worried about the smallest number of such primes in the "representation".