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Around 1955, Stanislaw Ulam identified the lucky number sequence, 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, …. First eliminate the even numbers. Then take 3; strike out every third number not yet eliminated. This eliminates 5, 11, 17, 23, and so on. The third surviving number is 7; strike out every seventh integer not yet eliminated: 19, 39, and so on. The integers that survive are shown highlighted in yellow in the sieve. Amazingly, luckies share many properties with primes; for instance, they have the same asymptotic density. Also, the number of twin primes—primes that differ by two—is close to the number of twin luckies. The Goldbach conjecture, which states that every even number greater than 2 is the sum of two primes, has an equivalent for luckies: every even number is the sum of two luckies. |
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