## Goldbach’s Other Conjecture

### June 7, 2016

Although there are other ways to do this, we solve the problem by generating the double-squares, then checking if the difference to the target number is prime. The sequence of double-squares is 0, 2, 8, 18, 32, …, so starting from 0 we walk through the sequence by adding 1*2, 3*2, 5*2, 7*2, …:

(define (goldbach n) (let loop ((dsq 0) (k 1)) (if (< n dsq) (list) (if (prime? (- n dsq)) (list (- n dsq) (sqrt (/ dsq 2))) (loop (+ dsq k k) (+ k 2))))))

Then, to find all the numbers that refute Goldbach’s other conjecture, we write:

> (do ((n 3 (+ n 2))) (#f) (let ((g (goldbach n))) (when (null? g) (display n) (newline)))) 5777 5993

That program quickly produces the two results you see above, then it will run for a long time producing no further results, since it is now conjectured that those are the only two counter-examples to the conjecture. See A060003 for more information, and especially the link given there to the paper by Hodges.

You can run the program at http://ideone.com/88RZja.

function Goldbach(x::Int64) # x is odd

P = primes(x)

for p in P

z = sqrt(x – p)

if z == round(z)

return true

end

end

return false

end

function main(n::Int64 = 1000)

for i = 1:n

x = 2 * round(Int64, 10000*rand()) + 1

if !Goldbach(x)

return x

end

end

return -1

end

It would be interesting to see how this is implemented in other languages.

A generator in Python.

June 7th, 2016.c:Output:On an Apple Power Mac G4 (AGP Graphics) (450MHz processor, 1GB memory) to run the solution took:

approximately twenty-two seconds on Mac OS 9.2.2 (International English) (the solution interpreted using Leonardo IDE 3.4.1);

approximately one second on Mac OS X 10.4.11 (the solution compiled using Xcode 2.2.1).

(I’m just trying to solve the problems posed by this ‘site whilst I try to get a job; I’m well aware that my solutions are far from the best – but, in my defence, I don’t have any traditional qualifications in computer science :/ )