Elementary Number Theory
... Proof. Assume p does not divide a. Then gcd(p, a) = 1 and hence, by Bézout’s Theorem, 1 = px + ay for suitable x and y. Multiplying by b we obtain b = pbx + aby. Since p divides ab we conclude p|b. Remark. Inductively we obtain from the theorem the slightly more general statement: Is p prime, p | ...
... Proof. Assume p does not divide a. Then gcd(p, a) = 1 and hence, by Bézout’s Theorem, 1 = px + ay for suitable x and y. Multiplying by b we obtain b = pbx + aby. Since p divides ab we conclude p|b. Remark. Inductively we obtain from the theorem the slightly more general statement: Is p prime, p | ...
1, N(3)
... largest prime P satisfying K S N - K < P _<_ N. It follows that P divides t and hence that n ? P for all n e S 2 . Hence all of the numbers in S2 lie between P and N . The number of numbers in S2 is thus S Z I <_ N - P <_ P5/8 < N518 <_ (log t) 3/a = O (log t/log log t), where, in obtaining the seco ...
... largest prime P satisfying K S N - K < P _<_ N. It follows that P divides t and hence that n ? P for all n e S 2 . Hence all of the numbers in S2 lie between P and N . The number of numbers in S2 is thus S Z I <_ N - P <_ P5/8 < N518 <_ (log t) 3/a = O (log t/log log t), where, in obtaining the seco ...
Math 784: algebraic NUMBER THEORY
... be the size of g − ah. Let b be an integer ≥ k1 . Associate the integer j=0 `n−j bj with an n-tuple (`1 , . . . , `n ). Show that the integer associated with (k1 , . . . , kn ) is greater than the integer associated with (k10 , . . . , kn0 ). Explain why (0, 0, . . . , 0) is obtained by continuing t ...
... be the size of g − ah. Let b be an integer ≥ k1 . Associate the integer j=0 `n−j bj with an n-tuple (`1 , . . . , `n ). Show that the integer associated with (k1 , . . . , kn ) is greater than the integer associated with (k10 , . . . , kn0 ). Explain why (0, 0, . . . , 0) is obtained by continuing t ...
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.