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# 239

The largest prime factor of *n*^{2} + 1 is at least 17, provided *n* is greater than the prime number 239.

In 1706, Machin discovered that π/4 = 4 arctan(1/5) - arctan(1/239). Shanks used this to calculate π to 707 places in 1873, but only the first 527 digits were correct.

A solution to *x*^{4} + 104^{4} = 58136^{2} + 1. [McLean]

*Magic Square Lexicon: Illustrated* by H. D. Heinz and J. R. Hendricks, includes 239 terms associated with magic squares, cubes, tesseracts, stars, etc.

The smallest prime factor of the palindrome 1234567654321. Note that 239 is the smallest prime with period length 7 (the palindrome's middle digit). [Yates]

239^{2} + 1 = 2 * 13^{4}. [Ljunggren]

A total cholesterol level of over 239 milligrams per deciliter is considered to be high.

The largest number that cannot be written as the sum of at most eight positive cubes: 239 = 5^3 + 3^3 + 3^3 + 3^3 + 2^3 + 2^3 + 2^3 + 2^3 + 1^3.

239 requires nineteen fourth powers to represent it. [Deshouillers]

How long does it take your computer to factor eighty factorial + 239? [Honaker]

The book *Pandolfini's Endgame Course: Basic Endgame Concepts Explained by America's Leading Chess Teacher* consists of 239 specific endgame positions, progressing from elementary endings to some subtle minor piece and pawn situations.

In a breeder reactor, uranium nuclei absorb neutrons to produce plutonium-239.

K. 239 (Serenata Notturna) is Mozart's only work for two orchestras. [Schroeppel]

HAKMEM, alternatively known as AI Memo 239, is a technical report of the MIT AI Lab that describes a wide variety of hacks, primarily useful and clever algorithms for mathematical computation.

Boland wrote 239#*2^239-1 is prime. Luhn found that 239#*2^239+/-1 are twins. [Luhn]

Professor Dick Solomon, lead character on the TV show *Third Rock from the Sun*, teaches his classes in room number 239. [Rupinski]

One saros cycle is almost exactly 239 anomalistic months. [McCranie]

In *The Simpsons* episode "Homer's Night Out," Homer
weighs himself at 239 pounds and resolves to exercise.

Saint Petersburg Lyceum 239 is a public high school in Russia that specializes in mathematics and physics. Among its famous alumni is Yuri Matiyasevich, who solved Hilbert's tenth problem.

Arctan(1/239) in degrees begins 0.239..., and this is the only positive integer for which this is true. [Wu and Caldwell]

239 is conjectured to be the largest number n such that n! can be written as the product of distinct factors in the range from n+1 to 2n, inclusive. [Chandler and Noe]

Only two positive integers may not be written as the sum of nine non-negative cubes: 23 and 239. [Dickson]

239 = 2*3*5*11 - 7*13, proof of primality. [Wood]

1+3+5+7+....+237+239 = 239+241+243+...+335+337. Note that 239 and 337 are both primes. [Meller]

239 is the 3rd NSW prime, following 7 and 41. Note that 7 + 41 + 239 = 7 * 41. [Menhinick]

The number of composite-numbered days of the month in both leap years and non-leap years. [Homewood]

The smallest prime in the only case of four 3-digit primes of forms, 4p+3, 6p+5, 8p+7, 10p+9, where p is a prime, (case p=59), i.e., 239, 359, 479, 599. Note that these forms have the general form kn+(k-1), where k is each of the first four even composite numbers, 4, 6, 8, 10. [Loungrides]

The first member of the smallest prime quadruple in 6p-1 progression: (239, 1433, 8597, 51581). [Bajpai]

The number of missing on board Malaysia Airlines Flight 370, which disappeared on 8 March 2014 while flying from Kuala Lumpur International Airport, Malaysia, to its destination, Beijing Capital International Airport in China.

The reversal of the 239th prime number (1499) is the 239th emirp (9941). [Gaydos]

The first known observation of Halley's Comet took place in 239 B.C., according to the European Space Agency.

The 52nd prime number (239) has a Collatz trajectory length of 52. [Gaydos]

The smallest integer greater than 1 that has the same Collatz trajectory length (52) as its square (57121). [Gaydos]