By Rudenskaya O.G.

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I am unaware of a rigorous probabilistic proof in the literature. Chapter 3 A One-Dimensional Probabilistic Packing Problem Consider n molecules lined up in a row. From among the n 1 nearest neighbor pairs, select one pair at random and “bond” the two molecules together. Now from all the remaining nearest neighbor pairs, select one pair at random and bond the two molecules together. Continue like this until no nearest neighbor pairs remain. Let MnI2 denote the random variable that counts the number of bonded molecules.

1 rD1 r log n exists; the limit is called Euler’s constant and is denoted by . One has 0:5772. For a proof, see, for example, [25]. Show that qkIk 1 1 k 1 e 2 ; as k ! t / D gj . r/ dr Show that since gj . / D j j t C O. s/e j C1 lim gj . 39) ! 0, one has D 1: On the other hand, since bj appears instead of bk lim ds; t 2 . ; 1/: 1 ds D D b, show that one also has 1: Rt You are Rinvited to show that the appropriate terms in gj . r/ dr ds cancel each other out and to obtain a finite limiting expression as !

16 2 Relatively Prime Pairs and Square-Free Numbers by a different prime pk are “independent” events, the “probability” that the two “randomly” selected positive Q integers1 are such that, for every k, at least one of them is not divisible by pk is 1 /. 1 pk2 two “randomly” selected positive integers are relatively prime. 1. For the probabilistic proof, the second alternative suggested in the second paragraph of the chapter will be more convenient. Thus, we choose an integer from Œn uniformly at random and then choose a second integer from Œn uniformly at random.