 Is there a pair of integers (a, b) such that Daniell Phillips 2022-01-07 Answered
Is there a pair of integers $$\displaystyle{\left({a},{b}\right)}$$ such that $$\displaystyle{a},{x}_{{1}},{y}_{{1}},{b}$$ is part of an arithmetic sequences and $$\displaystyle{a},{x}_{{2}},{y}_{{2}},{b}$$ is part of a geometric sequence with $$\displaystyle{x}_{{1}},{x}_{{2}},{y}_{{1}},{y}_{{2}}$$ all integers?

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It is given that $$\displaystyle{x}_{{1}},{x}_{{2}},{y}_{{1}},{y}_{{2}}$$ are all integers.
$$\displaystyle{a},{x}_{{1}},{y}_{{1}},{b}$$ is part of an arithmetic sequences.
Thus, $$\displaystyle{x}_{{1}}−{a}={y}_{{1}}−{x}_{{1}}$$ the common difference is same in arithmetic sequences.
$$\displaystyle{x}_{{1}}−{a}={y}_{{1}}−{x}_{{1}}$$
$$\displaystyle{2}{x}_{{1}}={y}_{{1}}+{a}$$
$$\displaystyle{a}={2}{x}_{{1}}−{y}_{{1}}$$
So, $$\displaystyle{a}$$ is integer as $$\displaystyle{x}_{{1}}$$ and $$\displaystyle{y}_{{1}}$$ are integers. (1)
$$\displaystyle{y}_{{1}}−{x}_{{1}}={b}−{y}_{{1}}$$ the common difference is same in arithmetic sequences.
$$\displaystyle{y}_{{1}}−{x}_{{1}}={b}−{y}_{{1}}$$
$$\displaystyle{2}{y}_{{1}}={x}_{{1}}+{b}$$
$$\displaystyle{b}={2}{y}_{{1}}−{x}_{{1}}$$
So, $$\displaystyle{b}$$ is integer as $$\displaystyle{x}_{{1}}$$ and $$\displaystyle{y}_{{1}}$$ are integers. (2)
It is given that $$\displaystyle{a},{x}_{{2}},{y}_{{2}},{b}$$ is part of a geometric sequence. Hence, $$\displaystyle{\frac{{{x}_{{2}}}}{{{a}}}}={\frac{{{y}_{{2}}}}{{{x}_{{2}}}}}$$ the common ratio is same in geometric sequence.
$$\displaystyle{x}_{{2}}{x}_{{2}}={a}{y}_{{2}}$$
$$\displaystyle{a}={\frac{{{{x}_{{2}}^{{{2}}}}}}{{{y}_{{2}}}}}$$
$$\displaystyle{x}_{{2}}=\sqrt{{{a}{y}_{{2}}}}$$
Similarly, $$y_2=\sqrt{bx_2}$$
Also, $$\displaystyle{\frac{{{x}_{{2}}}}{{{a}}}}={\frac{{{b}}}{{{y}_{{2}}}}}$$ the common ratio is same in geometric sequence.
$$\displaystyle{a}{b}={x}_{{2}}{y}_{{2}}$$
$$\displaystyle{a}{b}$$ is an integer as $$\displaystyle{x}_{{2}}$$ and $$\displaystyle{y}_{{2}}$$ are integerand the product of two integer is an integer.
Thus, from equations it is proved that there is a pair of integers $$\displaystyle{\left({a},{b}\right)}$$.