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Omni-MATH / (a) Does sum_{i=1}^{p-1} frac{1}{i} equiv 0left(bmod p^{2}right) for all odd prime numbers p? (Note that (1)/(i) denotes the number such that …

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(a) Does $\sum_{i=1}^{p-1} \frac{1}{i} \equiv 0\left(\bmod p^{2}\right)$ for all odd prime numbers pp? (Note that 1i\frac{1}{i} denotes the number such that $\left.i \cdot \frac{1}{i} \equiv 1\left(\bmod p^{2}\right)\right)$ (b) Do there exist 2017 positive perfect cubes that sum to a perfect cube? (c) Does there exist a right triangle with rational side lengths and area 5? (d) A magic square is a 3×33 \times 3 grid of numbers, all of whose rows, columns, and major diagonals sum to the same value. Does there exist a magic square whose entries are all prime numbers? (e) Is ∏pp2+1p2−1=22+122−1⋅32+132−1⋅52+152−1⋅72+172−1⋅…\prod_{p} \frac{p^{2}+1}{p^{2}-1}=\frac{2^{2}+1}{2^{2}-1} \cdot \frac{3^{2}+1}{3^{2}-1} \cdot \frac{5^{2}+1}{5^{2}-1} \cdot \frac{7^{2}+1}{7^{2}-1} \cdot \ldots a rational number? (f) Do there exist an infinite number of pairs of distinct integers (a,b)(a, b) such that aa and bb have the same set of prime divisors, and a+1a+1 and b+1b+1 also have the same set of prime divisors?
Plain-text mathematical notation (without MathML)
(a) Does $\sum_{i=1}^{p-1} \frac{1}{i} \equiv 0\left(\bmod p^{2}\right)$ for all odd prime numbers p? (Note that (1)/(i) denotes the number such that $\left.i \cdot \frac{1}{i} \equiv 1\left(\bmod p^{2}\right)\right)$ (b) Do there exist 2017 positive perfect cubes that sum to a perfect cube? (c) Does there exist a right triangle with rational side lengths and area 5? (d) A magic square is a 3×3 grid of numbers, all of whose rows, columns, and major diagonals sum to the same value. Does there exist a magic square whose entries are all prime numbers? (e) Is ∏_(p)(p²+1)/(p²−1)=(2²+1)/(2²−1)⋅(3²+1)/(3²−1)⋅(5²+1)/(5²−1)⋅(7²+1)/(7²−1)⋅… a rational number? (f) Do there exist an infinite number of pairs of distinct integers (a,b) such that a and b have the same set of prime divisors, and a+1 and b+1 also have the same set of prime divisors?
Original LaTeX notation
(a) Does $\sum_{i=1}^{p-1} \frac{1}{i} \equiv 0\left(\bmod p^{2}\right)$ for all odd prime numbers $p$? (Note that $\frac{1}{i}$ denotes the number such that $\left.i \cdot \frac{1}{i} \equiv 1\left(\bmod p^{2}\right)\right)$ (b) Do there exist 2017 positive perfect cubes that sum to a perfect cube? (c) Does there exist a right triangle with rational side lengths and area 5? (d) A magic square is a $3 \times 3$ grid of numbers, all of whose rows, columns, and major diagonals sum to the same value. Does there exist a magic square whose entries are all prime numbers? (e) Is $\prod_{p} \frac{p^{2}+1}{p^{2}-1}=\frac{2^{2}+1}{2^{2}-1} \cdot \frac{3^{2}+1}{3^{2}-1} \cdot \frac{5^{2}+1}{5^{2}-1} \cdot \frac{7^{2}+1}{7^{2}-1} \cdot \ldots$ a rational number? (f) Do there exist an infinite number of pairs of distinct integers $(a, b)$ such that $a$ and $b$ have the same set of prime divisors, and $a+1$ and $b+1$ also have the same set of prime divisors?

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Source and history

task_id
69d4b421-ce06-5997-9415-3d46a5338802
task_key
test--69d4b421-ce06-5997-9415-3d46a5338802
task_revision_id
1
config / split
/ test

Official source

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