Exercise 3.6.2 (Maximum value of $R(n)$ for positive $n \leq 1000$)

Question

What is the maximum value of $R(n)$ for positive $n \leq 1000$?

Richard Ganaye · Accepted Answer

\begin{proof} Note that $R(325) = R(5^2\cdot 13) = 4 \cdot 3 \cdot 2 = 24$. Here $5$ and $13$ are the smallest primes of the form $4k+1$. If we want to improve this score, we must increase the exponent of $5$ or $13$, but $5^3 \cdot 13>1000$ and a fortiori $5^2 \cdot 13^2 >1000$.

Thus the maximum value of $R(n)$ for positive $n \leq 1000$ if $24$.
\end{proof}

Check:

\begin{verbatim}
def R(n):
    prod = 1
    decomposition = True
    for p, alpha in factor(n):
        if (p % 4 == 3) and (alpha % 2 == 1):
            decomposition = False
            break
        if p % 4 == 1:
            prod  = prod * (alpha + 1)
    if decomposition:
        return 4 * prod
    else:
         return 0
         
nrec, maxi = 0, 0
for n in range(1,1001):
    rec = R(n)
    if rec > maxi:
        nrec = n
        maxi = rec
print(nrec, maxi)

(325, 24)
\end{verbatim}

Exercise 3.6.2 (Maximum value of $R(n)$ for positive $n \leq 1000$)

Answers

Comments

Exercise 3.6.2 (Maximum value of $R(n)$ for positive $n \leq 1000$)

Answers

Comments

Add answer