Exercise 3.6.1 (Four consecutive positive integers, each with the property that $r(n) = 0$)

Question

Find four consecutive positive integers, each with the property that $r(n) = 0$.

Richard Ganaye · Accepted Answer

\begin{proof} A possible answer is $$6, 7, 8, 9,$$ because $3 \mid 6$ and $3$ is a prime of the form $4k+3$, $7$ is also a  prime of the form $4k+ 3$, $8 = 2^3$, with an exponent greater than $2$, and $3 \mid 9$.

Longer sequences are given by
$$18, 19, 20, 21, 22 , 23, 24,$$
or
$$
42, 43, 44 , 45, 46, 47, 48, 49.
$$
\end{proof}

\bigskip
The following program, written in Sage, can be used.
\begin{verbatim}
def r(n):
    t = 0
    decomposition = True
    for p, alpha in factor(n):
        if (p == 2 and alpha >= 2) or (p % 4 == 3):
            decomposition = False
            break
        if p % 4 == 1:
            t += 1
    if decomposition:
        return 2^(t+2)
    else:
         return 0
\end{verbatim}

Exercise 3.6.1 (Four consecutive positive integers, each with the property that $r(n) = 0$)

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Exercise 3.6.1 (Four consecutive positive integers, each with the property that $r(n) = 0$)

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