(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 + 21^{2}

15 = 7 + 22^{2}

21 = 3 + 23^{2}

25 = 7 + 23^{2}

27 = 19 + 22^{2}

33 = 31 + 21^{2}

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.

^{?}View Code RSPLUS

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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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