ON FINITE SUMS OF RECIPROCALS OF DISTINCT
... formally distinct sums in Pr_χ are separated by a distance of more than σ and hence, each element π of Pr-± gives rise to a half-open interval [TΓ, π + σ) which is disjoint from any other interval [TΓ', TΓ' + σ) for TΓ Φ τr'eP r _!. Therefore Ac(S) = \Jnepr-.x\.π9π + σ) is the disjoint union of exac ...
... formally distinct sums in Pr_χ are separated by a distance of more than σ and hence, each element π of Pr-± gives rise to a half-open interval [TΓ, π + σ) which is disjoint from any other interval [TΓ', TΓ' + σ) for TΓ Φ τr'eP r _!. Therefore Ac(S) = \Jnepr-.x\.π9π + σ) is the disjoint union of exac ...
Pythagorean triples from fractions
... 2. Find the sum of their reciprocals, simplifying fractions where appropriate. 3. The numerator and the denominator give the first two numbers in your Pythagorean triple. Substitute these numbers into Pythagoras’ Theorem to find the third number. E.g. ...
... 2. Find the sum of their reciprocals, simplifying fractions where appropriate. 3. The numerator and the denominator give the first two numbers in your Pythagorean triple. Substitute these numbers into Pythagoras’ Theorem to find the third number. E.g. ...
1.4 Deductive Reasoning
... Inductive*reasoning*is*not*a*proof*of*anything*except*for*possibilities*that*you*tested.* There*could*always*be*a*counterexample*just*around*the*corner.* ...
... Inductive*reasoning*is*not*a*proof*of*anything*except*for*possibilities*that*you*tested.* There*could*always*be*a*counterexample*just*around*the*corner.* ...
NON-CONVERGING CONTINUED FRACTIONS RELATED TO THE
... On the other hand, it is easy to infer from Equations (2.4) and (2.5) that F and G are not rational functions. This follows from some easy considerations involving the degree of rational functions. This also follows from other simple observations as shown in [4] where the transcendence of both F and ...
... On the other hand, it is easy to infer from Equations (2.4) and (2.5) that F and G are not rational functions. This follows from some easy considerations involving the degree of rational functions. This also follows from other simple observations as shown in [4] where the transcendence of both F and ...
ELEMENTARY NUMBER THEORY
... demanded in the way of specific prerequisites. A significant portion of the book can be profitably read by anyone who has taken the equivalent of a first-year college course in mathematics. Those who have had additional courses will generally be better prepared, if only because of their enhanced mat ...
... demanded in the way of specific prerequisites. A significant portion of the book can be profitably read by anyone who has taken the equivalent of a first-year college course in mathematics. Those who have had additional courses will generally be better prepared, if only because of their enhanced mat ...
Remarks on number theory I
... an everywhere dense G,, in (l, v), i . e.. they are the countable intersection of dense open sets . But it is not difficult to show that they have measure 0, in fact they must be Liouville numbers (a number y is called a Liouville number if y is irrational and l y-a /b i < I /V is solvable in intege ...
... an everywhere dense G,, in (l, v), i . e.. they are the countable intersection of dense open sets . But it is not difficult to show that they have measure 0, in fact they must be Liouville numbers (a number y is called a Liouville number if y is irrational and l y-a /b i < I /V is solvable in intege ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".