(This article was first published on YGC, and kindly contributed to R-bloggers)
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
9 = 7 + 2
12
15 = 7 + 222
21 = 3 + 232
25 = 7 + 232
27 = 19 + 222
33 = 31 + 212
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
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Referring to http://learning.physics.iastate.edu/hodges/mm-1.pdf, this problem is very famous.
Using brute-force is the solution I can only think of. Surprisingly, it turns out very fast.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | require(gmp) n <- 1:10000 p <- n[as.logical(isprime(n))] for (i in seq(3,10000,2)) { if (any(p==i)) next x <- sqrt((i-p[p<i])/2) if (any(round(x) == x)) { next } else { cat (i, "\n") } } |
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