239 (number)

239 is a prime number. The next is 241, with which it forms a pair of twin primes; hence, it is also a Chen prime. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1 (with no exponentiation implied). 239 is a factor of the repdigit 1111111, with the other prime factor being 4649. 239 is also a happy number. 239 is the smallest positive integer d such that the imaginary quadratic field Q() has class number = 15. 239 is the smallest number that contains the highest possible digit in all bases from 2 to 12: • 111011112 • 222123 • 32334 • 14245 • 10356 • 4617 • 3578 • 2859 • 23910 • 1A811 • 17B12 The next number with this property is 5927. ==HAKMEM entry==

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. • 239/169 is a convergent of the simple continued fraction of the square root of 2, so that 2392 = 2 · 1692 − 1. • Related to the above, = 45°. • 239 · 4649 = 1111111, so 1/239 = 0.0041841 repeating, with period 7. • 239 can be written as bn − bm − 1 for b = 2, 3, and 4, a fact evidenced by its binary representation 11101111, ternary representation 22212, and quaternary representation 3233. • There are 239 primes 2 + 1 = 2x4 in positive integers are (x, y) = (1, 1) or (13, 239). ==References==