What is the least positive integer $n$ such that $(5n^4+3)$ is a prime number?

The answer is 2.

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Positive integers start from $1,2,3,4,5,\ldots$

For $n = 1,$

$5n^4 + 3$

$= 5(1)^4 +3$

$\space \space ~~~~~~ ~~~~~~~~ ~~~~~ = 8 \longrightarrow \color{#D61F06}\text{Not a prime number}.$

For $n =2,$

$5n^4+3$

$= 5(2)^4 + 3$

$\space \space ~~~~~~ ~~~~~~~~ ~~~~~= 83 \longrightarrow \color{#20A900}\text{Prime number}$

Hence $n = \boxed{2}$ is the least integer.