239

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Number

$239$ (two hundred and thirty-nine) is:

The $52$nd prime number

The $2$nd of the only two positive integers needing as many as $9$ positive cubes to express it:
$239 = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3$
The other is $23$.

The $3$rd positive integer after $79$, $159$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$

The $1$st of the $17$th pair of twin primes, with $241$

The $17$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$, $233$:
$2 \times 239 + 1 = 479$, which is prime.

The $39$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $188$, $190$, $192$, $193$, $203$, $208$, $219$, $226$, $230$, $236$:
$239 \to 2^2 + 3^2 + 9^2 = 4 + 9 + 81 = 94 \to 9^2 + 4^2 = 81 + 16 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$

The largest integer such that the largest prime factor of $n^2 + 1$ is less than $17$:
$239^2 + 1 = 57122 = 2 \times 13^4$

With $y = 13$, the only $x$ which is the solution of the indeterminate Diophantine equation $x^2 + 1 = 2 y^4$:
$239^2 + 1 = 2 \times 13^4$