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Omni-MATH / Find all positive integers n≥2 for which there exist n real numbers a₁<⋯<a_(n) and a real number r>0 such that the (1)/(2)n(n−1) differences a_(j)−a_(i) for 1≤i<j≤n are equal, in s…

Problem

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problem

Find all positive integers n≥2n \geqslant 2 for which there exist nn real numbers a1<⋯<ana_1<\cdots<a_n and a real number r>0r>0 such that the 12n(n−1)\tfrac{1}{2}n(n-1) differences aj−aia_j-a_i for 1≤i<j≤n1 \leqslant i<j \leqslant n are equal, in some order, to the numbers r1,r2,…,r12n(n−1)r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}.
Plain-text mathematical notation (without MathML)
Find all positive integers n≥2 for which there exist n real numbers a₁<⋯<a_(n) and a real number r>0 such that the (1)/(2)n(n−1) differences a_(j)−a_(i) for 1≤i<j≤n are equal, in some order, to the numbers r¹,r²,…,r^((1)/(2)n(n−1)).
Original LaTeX notation
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

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Source and history

task_id
ebc17199-82d2-5ee9-804f-31fd5d3842f2
task_key
test--ebc17199-82d2-5ee9-804f-31fd5d3842f2
task_revision_id
1
config / split
/ test

Official source

initial import