MathDB

Let

\begin{array}{cccc}{a_{1,1}} & {a_{1,2}} & {a_{1,3}} & {\dots} \\ {a_{2,1}} & {a_{2,2}} & {a_{2,3}} & {\cdots} \\ {a_{3,1}} & {a_{3,2}} & {a_{3,3}} & {\cdots} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots}\end{array}

be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that

a_{m, n}>m n

for some pair of positive integers

(m, n) .

B3