On Rough and Smooth Neighbors
... We claim that q ∈ Qp . Indeed, if n ≥ 1 is an integer for which P + (n) = p, property (i) implies that n is a quadratic residue modulo q. But then the equation P − (n + 1) = q is not possible, for otherwise n ≡ −1 (mod q) is a quadratic Q nonresidue by (ii). To construct examples of such primes q, l ...
... We claim that q ∈ Qp . Indeed, if n ≥ 1 is an integer for which P + (n) = p, property (i) implies that n is a quadratic residue modulo q. But then the equation P − (n + 1) = q is not possible, for otherwise n ≡ −1 (mod q) is a quadratic Q nonresidue by (ii). To construct examples of such primes q, l ...
Math 259: Introduction to Analytic Number Theory Elementary
... Even a piece of mathematics as venerable as Euclid’s proof of the infinitude of primes can continue to suggest Qn very difficult problems. For instance, let pn be the n-th prime and Pn = k=1 pk as before. We know that Pn + 1 must contain a new prime factor, which cannot be pn+1 once n > 1 (if only b ...
... Even a piece of mathematics as venerable as Euclid’s proof of the infinitude of primes can continue to suggest Qn very difficult problems. For instance, let pn be the n-th prime and Pn = k=1 pk as before. We know that Pn + 1 must contain a new prime factor, which cannot be pn+1 once n > 1 (if only b ...
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... We can also prove the latter by proving its contrapositive, i.e., we can prove if n is odd then 7n + 4 is odd. Since n is odd we have n = 2k + 1, for some integerk. Thus we have 7n + 4 = 7(2k + 1) + 4 = 14k + 10 + 1 = 2(7k + 5) + 1 = 2k′ + 1, where k′ = 7k + 5 is an integer. Example 2. ...
... We can also prove the latter by proving its contrapositive, i.e., we can prove if n is odd then 7n + 4 is odd. Since n is odd we have n = 2k + 1, for some integerk. Thus we have 7n + 4 = 7(2k + 1) + 4 = 14k + 10 + 1 = 2(7k + 5) + 1 = 2k′ + 1, where k′ = 7k + 5 is an integer. Example 2. ...
Chapter 3 Elementary Number Theory The expression lcm(m,n
... The expression lcm(m,n) stands for the least integer which is a multiple of both the integers m and n. The expression gcd(m,n) stands for the biggest integer that divides both m and n. Find lcm and gcd in the TI-85 CATALOG and place them into your custom catalog. Following the procedures of the firs ...
... The expression lcm(m,n) stands for the least integer which is a multiple of both the integers m and n. The expression gcd(m,n) stands for the biggest integer that divides both m and n. Find lcm and gcd in the TI-85 CATALOG and place them into your custom catalog. Following the procedures of the firs ...
Lesson Plan Template - Trousdale County Schools
... Tennessee State Standard(s) to be taught: (Write the entire standard) ...
... Tennessee State Standard(s) to be taught: (Write the entire standard) ...
Congruence Properties of the Function that Counts Compositions
... Suppose we know the sequence ϑ(n) modulo 2N . In this case the recurrences (5), (6) and the above fact show us that the sequence ϑ(n) is completely describable modulo 2N +1 as well. Further, note that Table 1 lists only those even and odd numbers n such that s2 (n + 2) ≤ 3. The recurrence (5) shows ...
... Suppose we know the sequence ϑ(n) modulo 2N . In this case the recurrences (5), (6) and the above fact show us that the sequence ϑ(n) is completely describable modulo 2N +1 as well. Further, note that Table 1 lists only those even and odd numbers n such that s2 (n + 2) ≤ 3. The recurrence (5) shows ...
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