For $(5,5)$: $k - y = 5$, $k + y = 5$ → $k = 5$, $y = 0$ → $x = 10$.

February 22, 2026

For $(5,5)$: $k - y = 5$, $k + y = 5$ → $k = 5$, $y = 0$ → $x = 10$.

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Question: How many lattice points lie on the hyperbola $x^2 - 4y^2 = 100$?
Solution: Rewrite as $x^2 = 4y^2 + 100$. Let $x = 2k$, then $4k^2 = 4y^2 + 100 \implies k^2 - y^2 = 25$. Factor as $(k - y)(k + y) = 25$. The factor pairs of 25 are $(1,25), (5,5), (-1,-25), (-5,-5)$. Solving:
For $(1,25)$: $k - y = 1$, $k + y = 25$ → $k = 13$, $y = 12$ → $x = 26$.
Negative pairs yield symmetric solutions. Total lattice points: $(\pm26, \pm12)$ and $(\pm10, 0)$, totaling $\boxed{6}$.Question: A mechanical engineer is designing a gear system where the torque output $ T(n) $ at segment $ n $ is defined recursively by $ T(n) = 2T(n-1) + 3 $, with $ T(1) = 4 $. If the system evaluates $ T(m) $ for a specific $ m $, and the sum $ S(m) = \sum_{n=1}^{m} T(n) $ models total energy over time, compute $ S(5) $.
Solution: We are given a recurrence:
T(1) = 4, \quad T(n) = 2T(n-1) + 3 \text{ for } n \geq 2.

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