Packet 10: Tossup 9

The probability that a number less than or equal to N is one of these numbers is approximately one over the natural log (15[1])of N. (15[3])The letter pi represents a function which counts how many of these numbers are between (15[1])one (15[1])and x. (15[1])Arbitrarily long (-5[1])arithmetic (-5[1])sequences (15[1])can be constructed within these numbers according to the Green-Tao theorem. (15[1])Multiplying several of these numbers and then (*) adding 1 can generate another one, a key step in proving (10[1])that there are infinitely many (10[1])of these numbers. (10[1]-5[1])It remains unproven that there are infinitely many “twin” pairs of these numbers, which differ by two. For 10 points, name these numbers which are only divisible by one and themselves. (10[1])■END■ (10[5])

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