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Home » \[ h_0 - r(4)^2 = h_0 - 16 \] - jntua results

\[ h_0 - r(4)^2 = h_0 - 16 \] - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 To see if 30 is the largest such integer, observe:

📌 One of these integers is divisible by 5.

📌 At least one is divisible by 2, and at least one by 4 (since there are five numbers), ensuring divisibility by \( 2^3 = 8 \). However, the product is already guaranteed by \( 2 \times 3 = 6 \), and typically one is divisible by 4 but not consistently ensuring a higher power.

📌 But more carefully: among five consecutive numbers, the minimal guaranteed prime power factorizations are:

📌 One multiple of 5 → factor of 5

📌 At least two multiples of 2 (possibly one of which is a multiple of 4), but minimal guaranteed is one factor of 4 and one of 2 → ensures \( 2^3 \) is not guaranteed (e.g., sequence 1–5: 2 and 4 → \( 2 \times 4 = 8 \), divisible by 8? 2 and 4 give \( 2^3 \)).

📌 Wait: 2 and 4 → \( 2 \times 4 = 8 \), divisible by 8. Another example: 3–7 → 4 → divisible by 4, 6 → divisible by 2, so again 8.

📌 But sequence 4–8: 4 (div by 4), 6 (div by 2), 8 (div by 8) → divisible by 8.

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