Content deleted Content added
Citation bot (talk | contribs) Removed URL that duplicated identifier. Removed parameters. | Use this bot. Report bugs. | Suggested by Headbomb | Linked from Wikipedia:WikiProject_Academic_Journals/Journals_cited_by_Wikipedia/Sandbox | #UCB_webform_linked 690/1032 |
|||
| (2 intermediate revisions by 2 users not shown) | |||
Line 9:
:<math> \left | \alpha -\frac{p}{q} \right | < \frac{1}{q^2} </math>
is satisfied by infinitely many integers ''p'' and ''q''. This shows that any irrational number has [[irrationality measure#irrationality_exponent|irrationality exponent]] at least 2.
The [[Thue–Siegel–Roth theorem]] says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2. The Thue–Siegel–Roth theorem uses advanced techniques of number theory, but many simpler numbers such as the [[golden ratio]] <math>(1+\sqrt{5})/2</math> can be much more easily verified to be inapproximable beyond exponent 2.
Line 58:
{{collapse bottom}}
This theorem forms the basis for [[Wiener's attack]], a polynomial-time exploit of the [[RSA (cryptosystem)|RSA cryptographic protocol]] that can occur for an injudicious choice of public and private keys (specifically, this attack succeeds if the prime factors of the public key ''n'' = ''pq'' satisfy ''p'' < ''q'' < 2''p'' and the private key ''d'' is less than (1/3)''n''<sup>1/4</sup>).<ref>{{cite journal|last=Wiener|first=Michael J.|date=1990|title=Cryptanalysis of short RSA secret exponents
==See also==
| |||