Question 5
Let $p>3$ be a prime. In each part, $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$, and the congruence condition belongs to that part. Calculate:
(a) If $p\equiv1\pmod4$,
$$S=\sum_{k=1}^{(p-1)/2}\left(\left\lfloor\frac{2k^2}{p}\right\rfloor-2\left\lfloor\frac{k^2}{p}\right\rfloor\right).$$
(b) If $p\equiv1\pmod8$,
$$S=\sum_{k=1}^{(p-1)/2}\left\lfloor\frac{k^2}{p}\right\rfloor.$$
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
(a) If $p\equiv1\pmod4$,
$$S=\sum_{k=1}^{(p-1)/2}\left(\left\lfloor\frac{2k^2}{p}\right\rfloor-2\left\lfloor\frac{k^2}{p}\right\rfloor\right).$$
(b) If $p\equiv1\pmod8$,
$$S=\sum_{k=1}^{(p-1)/2}\left\lfloor\frac{k^2}{p}\right\rfloor.$$
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.