ProjectEuler-Problem 46

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 + 2x1²
15 = 7 + 2x2²
21 = 3 + 2x3²
25 = 7 + 2x3²
27 = 19 + 2x2²
33 = 31 + 2x1²

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.

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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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