Project Euler Problem 586
The number 209 can be expressed as a^2 + 3ab + b^2 in two distinct ways: qquad 209 = 8^2 + 3cdot 8cdot 5 + 5^2 qquad 209
Solution
Answer: 82490213
A standard way to attack this problem is to reinterpret
$$k=a^2+3ab+b^2$$
as a norm form in a quadratic integer ring. Specifically,
$$a^2+3ab+b^2=(a+\alpha b)(a+\beta b)$$
where $\alpha,\beta$ are roots of $x^2-3x+1=0$. This places the problem in the arithmetic of the real quadratic field $\mathbb{Q}(\sqrt{5})$, where counting representations becomes a divisor-counting problem with local constraints.
After converting the representation count into multiplicative data on the prime factorization of $k$, one finds that the number of admissible representations is governed by the exponents of primes that split in the quadratic field (equivalently, primes satisfying a congruence condition mod $5$), together with corrections for ordering ($a>b>0$) and exceptional units. The task reduces to counting integers $k\le 10^{15}$ whose induced representation count is exactly $40$.
Using a recursive multiplicative enumeration over valid prime-exponent signatures (pruned by the bound $10^{15}$), and validating against the provided checkpoints:
- $f(10^5,4)=237$
- $f(10^8,6)=59517$
the computation yields the final value below.
Answer: 82490213