Project Euler Problem 46: Goldbach's other conjecture

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

$$ 9 = 7 + 2 \cdot 1^2 \\ 15 = 7 + 2 \cdot 2^2 \\ 21 = 3 + 2 \cdot 3^2 \\ 25 = 7 + 2 \cdot 3^2 \\ 27 = 19 + 2 \cdot 2^2 \\ 33 = 31 + 2 \cdot 1^2 $$

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?

Version 1: Brute force

Formally stated, Goldbach's Other Conjecture says that all odd composite numbers $n$ can be expressed as

$$ n = 2k^2 + p $$

for some integer $k$ and prime number $p$. Let

$$ S_n = \{ n - 2k^2 : k = 1, 2, \dotsc, \lfloor \sqrt{\frac{n}{2}} \rfloor \} $$

If any element $n - 2k^2$ of $S_n$ is prime, then we call $k$ a witness to Goldbach's Other Conjecture for $n$. We are asked to find the smallest $n$ that has no witness, i.e. such that all elements of $S_n$ are composite.

Let $P_n$ be the set of all prime numbers stricly smaller than $n$, then our algorithm searches for the smallest $n$ such that $P_n \cap S_n = \emptyset$, providing a counterexample to Goldbach's Other Conjecture.

In [1]:
from IPython.display import Math, display

from collections import defaultdict
from itertools import count

We modify and augment a very space-efficient implementation (due to David Eppstein, UC Irvine) of the Sieve of Erastothenes to generate composite numbers and keep track of all prime numbers below it. In the outermost-loop below, primes will always be the list of all prime number strictly below $n$, i.e. it is equivalent to $P_n$. Note that if $n$ is odd and composite, then the largest prime below it is no greater than $n-2$, since $n-1$ is even. Since $\max S_n = n - 2$ searching through $P_n$ is sufficient (i.e. if $n-2$ is prime, then $n-2 \in P_n$.) The loop terminates when we encounter an $n$ with no witnesses.

In [2]:
factors = defaultdict(list)
witness = {}
primes = set()
for n in count(2):
    if factors[n]:
        # n is composite
        for m in factors.pop(n):
            factors[n+m].append(m)
        if n % 2:
            # n is odd and composite
            for k in range(1, int((n/2)**.5)+1):
                p = n - 2*k**2
                if p in primes: 
                    # TODO: `in` is O(len(primes))
                    # could optimize by using set
                    witness[n] = k
                    break
            if not n in witness:
                break
    else:
        factors[n*n].append(n)
        primes.add(n)
In [3]:
n
Out[3]:
5777

The answer is 5777.

Note that not only is this implementation space-efficient, it is also very time efficient, since we incrementally build our list of primes, composites and witnesses incrementally from bottom-up. If we hadn't augmented the implementation of the prime sieve, we would have had to use the prime sieve to obtain all odd composites, and perform a primality test on all elements of $S_n$, which would (usually) require a prime factorization algorithm, which in turn (usually) requires a prime sieve, not to mention the fact that there would be many overlapping subproblems, i.e. many $n

Let's list out the witnesses to the first 100 numbers.

In [4]:
lines = [r'{0} &= {1} + 2 \cdot {2}^2 \\'.format(n, n - 2*witness[n]**2, witness[n]) for n in sorted(witness)]
In [5]:
Math(r"""
     \begin{{align}}
     {body}
     \end{{align}}
     """.format(body='\n'.join(lines[:100])))
Out[5]:
$$ \begin{align} 9 &= 7 + 2 \cdot 1^2 \\ 15 &= 13 + 2 \cdot 1^2 \\ 21 &= 19 + 2 \cdot 1^2 \\ 25 &= 23 + 2 \cdot 1^2 \\ 27 &= 19 + 2 \cdot 2^2 \\ 33 &= 31 + 2 \cdot 1^2 \\ 35 &= 17 + 2 \cdot 3^2 \\ 39 &= 37 + 2 \cdot 1^2 \\ 45 &= 43 + 2 \cdot 1^2 \\ 49 &= 47 + 2 \cdot 1^2 \\ 51 &= 43 + 2 \cdot 2^2 \\ 55 &= 53 + 2 \cdot 1^2 \\ 57 &= 7 + 2 \cdot 5^2 \\ 63 &= 61 + 2 \cdot 1^2 \\ 65 &= 47 + 2 \cdot 3^2 \\ 69 &= 67 + 2 \cdot 1^2 \\ 75 &= 73 + 2 \cdot 1^2 \\ 77 &= 59 + 2 \cdot 3^2 \\ 81 &= 79 + 2 \cdot 1^2 \\ 85 &= 83 + 2 \cdot 1^2 \\ 87 &= 79 + 2 \cdot 2^2 \\ 91 &= 89 + 2 \cdot 1^2 \\ 93 &= 61 + 2 \cdot 4^2 \\ 95 &= 23 + 2 \cdot 6^2 \\ 99 &= 97 + 2 \cdot 1^2 \\ 105 &= 103 + 2 \cdot 1^2 \\ 111 &= 109 + 2 \cdot 1^2 \\ 115 &= 113 + 2 \cdot 1^2 \\ 117 &= 109 + 2 \cdot 2^2 \\ 119 &= 101 + 2 \cdot 3^2 \\ 121 &= 113 + 2 \cdot 2^2 \\ 123 &= 73 + 2 \cdot 5^2 \\ 125 &= 107 + 2 \cdot 3^2 \\ 129 &= 127 + 2 \cdot 1^2 \\ 133 &= 131 + 2 \cdot 1^2 \\ 135 &= 127 + 2 \cdot 2^2 \\ 141 &= 139 + 2 \cdot 1^2 \\ 143 &= 71 + 2 \cdot 6^2 \\ 145 &= 137 + 2 \cdot 2^2 \\ 147 &= 139 + 2 \cdot 2^2 \\ 153 &= 151 + 2 \cdot 1^2 \\ 155 &= 137 + 2 \cdot 3^2 \\ 159 &= 157 + 2 \cdot 1^2 \\ 161 &= 89 + 2 \cdot 6^2 \\ 165 &= 163 + 2 \cdot 1^2 \\ 169 &= 167 + 2 \cdot 1^2 \\ 171 &= 163 + 2 \cdot 2^2 \\ 175 &= 173 + 2 \cdot 1^2 \\ 177 &= 127 + 2 \cdot 5^2 \\ 183 &= 181 + 2 \cdot 1^2 \\ 185 &= 167 + 2 \cdot 3^2 \\ 187 &= 179 + 2 \cdot 2^2 \\ 189 &= 181 + 2 \cdot 2^2 \\ 195 &= 193 + 2 \cdot 1^2 \\ 201 &= 199 + 2 \cdot 1^2 \\ 203 &= 131 + 2 \cdot 6^2 \\ 205 &= 197 + 2 \cdot 2^2 \\ 207 &= 199 + 2 \cdot 2^2 \\ 209 &= 191 + 2 \cdot 3^2 \\ 213 &= 211 + 2 \cdot 1^2 \\ 215 &= 197 + 2 \cdot 3^2 \\ 217 &= 199 + 2 \cdot 3^2 \\ 219 &= 211 + 2 \cdot 2^2 \\ 221 &= 149 + 2 \cdot 6^2 \\ 225 &= 223 + 2 \cdot 1^2 \\ 231 &= 229 + 2 \cdot 1^2 \\ 235 &= 233 + 2 \cdot 1^2 \\ 237 &= 229 + 2 \cdot 2^2 \\ 243 &= 241 + 2 \cdot 1^2 \\ 245 &= 227 + 2 \cdot 3^2 \\ 247 &= 239 + 2 \cdot 2^2 \\ 249 &= 241 + 2 \cdot 2^2 \\ 253 &= 251 + 2 \cdot 1^2 \\ 255 &= 223 + 2 \cdot 4^2 \\ 259 &= 257 + 2 \cdot 1^2 \\ 261 &= 229 + 2 \cdot 4^2 \\ 265 &= 263 + 2 \cdot 1^2 \\ 267 &= 139 + 2 \cdot 8^2 \\ 273 &= 271 + 2 \cdot 1^2 \\ 275 &= 257 + 2 \cdot 3^2 \\ 279 &= 277 + 2 \cdot 1^2 \\ 285 &= 283 + 2 \cdot 1^2 \\ 287 &= 269 + 2 \cdot 3^2 \\ 289 &= 281 + 2 \cdot 2^2 \\ 291 &= 283 + 2 \cdot 2^2 \\ 295 &= 293 + 2 \cdot 1^2 \\ 297 &= 199 + 2 \cdot 7^2 \\ 299 &= 281 + 2 \cdot 3^2 \\ 301 &= 293 + 2 \cdot 2^2 \\ 303 &= 271 + 2 \cdot 4^2 \\ 305 &= 233 + 2 \cdot 6^2 \\ 309 &= 307 + 2 \cdot 1^2 \\ 315 &= 313 + 2 \cdot 1^2 \\ 319 &= 317 + 2 \cdot 1^2 \\ 321 &= 313 + 2 \cdot 2^2 \\ 323 &= 251 + 2 \cdot 6^2 \\ 325 &= 317 + 2 \cdot 2^2 \\ 327 &= 277 + 2 \cdot 5^2 \\ 329 &= 311 + 2 \cdot 3^2 \\ 333 &= 331 + 2 \cdot 1^2 \\ \end{align} $$

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