Prime Numbers
Share Embed Flag
Download ePaper

Prime Numbers

Prime Numbers Prime Numbers

thales.doa.fmph.uniba.sk
from thales.doa.fmph.uniba.sk More from this publisher
10.12.2012 Views

62 Chapter 1 PRIMES! so that A3(x) is the number of 3-term arithmetic progressions p

1.5 Exercises 63 (2) Assume outright (here is where we circumvent a great deal of hard work!) the fact that R2(n) 2 x 1, where A2 is a constant. This result can be derived via such sophisticated techniques as the Selberg and Brun sieves [Nathanson 1996]. (3) Use (1), (2), and the Cauchy–Schwarz inequality x 2 x ≤ x n=1 anbn (valid for arbitrary real numbers an,bn) to prove that for some positive constant A3 we have n=1 a 2 n n=1 #{n ≤ x : R2(n) > 0} >A3x, for all sufficiently large values of x, this kind of estimate being what is meant by “positive lower density” for the set S. (Hint: Define an = R2(n) and (bn) to be an appropriate binary sequence.) As discussed in the text, Shnirel’man proved that this lower bound on density implies his celebrated result that for some fixed s, every integer starting with 2 is the sum of at most s primes. It is intriguing that an upper bound on Goldbach representations—as in task (2)—is the key to this whole line of reasoning! That is because, of course, such an upper bound reveals that representation counts are kept “under control,” meaning “spread around” such that a sufficient fraction of even n have representations. (See Exercise 9.80 for further applications of this basic bounding technique.) 1.45. Assuming the prime k-tuples Conjecture, 1.2.1 show that for each k there is an arithmetic progression of k consecutive primes. 1.46. Note that each of the Mersenne primes 2 2 − 1, 2 3 − 1, 2 5 − 1isa member of a pair of twin primes. Do any other of the known Mersenne primes from Table 1.2 enjoy this property? 1.47. Let q be a Sophie Germain prime, meaning that s =2q + 1 is likewise prime. Prove that if also q ≡ 3(mod4)andq>3, then the Mersenne number Mq =2 q −1 is composite, in fact divisible by s. A large Sophie Germain prime is Kerchner and Gallot’s q = 18458709 · 2 32611 − 1, with 2q + 1 also prime, so that the resulting Mersenne number Mq is a truly gargantuan composite of nearly 10 104 decimal digits. 1.48. Prove the following relation between Mersenne numbers: gcd(2 a − 1, 2 b − 1) = 2 gcd(a,b) − 1. b 2 n

62 Chapter 1 PRIMES!<br />

so that A3(x) is the number of 3-term arithmetic progressions p

logo

products

Resources

Company

Terms of service
Privacy policy
Cookie policy
Cookie settings
Imprint
Change language
Made with love in Switzerland
© 2024 Yumpu.com all rights reserved

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!