239 (number)

Two hundred thirty-nine is the natural number following 238 and preceding 240.

Mathematics

It is a prime number. The next is 241, with which it comprises a pair of twin primes. 239 is a Sophie Germain prime and a Newman-Shanks-Williams prime. HAKMEM (incidentally AI memo 239 of the MIT AI Lab) included an item on the properties of 239, including these:

When expressing 239 as a sum of square numbers, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positive cubes (23 is the only other such integer), and the maximum number (19) of fourth powers.
239/(13)² is a convergent to the continued fraction of the square root of two, so that 239² = 2 · 13⁴ - 1.
Related to the above, π/4 = 4 arctan(1/5) - arctan(1/239).
239 · 4649 = 1111111, so 1/239 = 0.0041841 repeating, with period 7.
239 can be written as bⁿ - b^m - 1 for b = 2, 3, and 4, a fact evidenced by its binary representation 11101111, ternary representation 22212, and quaternary representation 3233.

K239 is Mozart's only work for two orchestras.
In The Simpsons episode "Homer's Night Out", Homer weighs himself and resolves to exercise. Six months later he weighs himself again and again resolves to exercise. Both times he weighed exactly 239 pounds.
The years A.D. 239 and 239 BC.

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