Pell`s equation and units in real quadratic fields
... \The smallest non-trivial integer solution of x2 94y2 = 1 is x = 2143295 and y = 221064:" This intrigued me, for some reason, and I asked what some other solutions were. I was urged to pick a number for D, and I picked 151. Solving these equations using a somewhat-better-than-brute-force method took ...
... \The smallest non-trivial integer solution of x2 94y2 = 1 is x = 2143295 and y = 221064:" This intrigued me, for some reason, and I asked what some other solutions were. I was urged to pick a number for D, and I picked 151. Solving these equations using a somewhat-better-than-brute-force method took ...
Chapter Two
... The most basic use of limits is to describe how a function behavior as the independent variable approaches a given value. ...
... The most basic use of limits is to describe how a function behavior as the independent variable approaches a given value. ...
Elementary Number Theory Definitions and Theorems
... for proof problems. The definitions given here (e.g., of divisibility) are the “authoritative” definitions, and you should use those definitions in proofs. The results stated here are those you are free to use and refer to in proofs; in general, anything else (e.g., a theorem you might have learned ...
... for proof problems. The definitions given here (e.g., of divisibility) are the “authoritative” definitions, and you should use those definitions in proofs. The results stated here are those you are free to use and refer to in proofs; in general, anything else (e.g., a theorem you might have learned ...
Regularity of minimizers of the area functional in metric spaces
... the last term depends only on f , we find that for every f ∈ L1 (∂Ω) there is a minimizer in BV(Ω) of the integral Z p Z ...
... the last term depends only on f , we find that for every f ∈ L1 (∂Ω) there is a minimizer in BV(Ω) of the integral Z p Z ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.