Math 5330 Spring 2016 Exam 1 Solutions In class questions 1. (10
... Solution: This is a special case of a result from class. If d “ gcdpm, nq then divides both m and n so it divides m ´ n. Consequently, d is a common divisor of m and m ´ n, giving gcdpm, nq ď gcdpm, m ´ nq. But if D “ gcdpm, m ´ nq, then D divides m and m ´ n, so D divides their difference, m ´ pm ´ ...
... Solution: This is a special case of a result from class. If d “ gcdpm, nq then divides both m and n so it divides m ´ n. Consequently, d is a common divisor of m and m ´ n, giving gcdpm, nq ď gcdpm, m ´ nq. But if D “ gcdpm, m ´ nq, then D divides m and m ´ n, so D divides their difference, m ´ pm ´ ...
Irrational square roots
... The proof is correct! Your student has earned a perfect score instead of 0, but did not learn what you wanted to teach. Next time you are teaching this course, you do not repeat your mistake! You ask yourself: for what prime numbers does this “even-odd” proof work? You easily see that it works for a ...
... The proof is correct! Your student has earned a perfect score instead of 0, but did not learn what you wanted to teach. Next time you are teaching this course, you do not repeat your mistake! You ask yourself: for what prime numbers does this “even-odd” proof work? You easily see that it works for a ...
Pythagorean Treasury Powerpoint - 8.1 ~ A collection of teaching
... There is no intuitive feeling that such an intimate connection exists between right angles and sums of squares. The existence of such a relationship is completely unexpected. The theorem establishes the truth of what is quite simply, an extremely odd fact. ...
... There is no intuitive feeling that such an intimate connection exists between right angles and sums of squares. The existence of such a relationship is completely unexpected. The theorem establishes the truth of what is quite simply, an extremely odd fact. ...
![pythagoreantreasury[1]](http://s1.studyres.com/store/data/008460234_1-b0fb1826394697cba2511ff556bba420-300x300.png)
Max Lewis Dept. of Mathematics, University of Queensland, St Lucia
... Germain primes and so on. Dickson’s conjecture is itself a special case of Schinzel’s Hypothesis H [18] concerning prime values of arbitrary polynomials. If k is also prime then the converse of Lemma 4 holds. Lemma 5. If p < q are odd primes and k(pq) = k where k is prime, then q = k(p 1) + 1. Proof ...
... Germain primes and so on. Dickson’s conjecture is itself a special case of Schinzel’s Hypothesis H [18] concerning prime values of arbitrary polynomials. If k is also prime then the converse of Lemma 4 holds. Lemma 5. If p < q are odd primes and k(pq) = k where k is prime, then q = k(p 1) + 1. Proof ...
Approximation to real numbers by algebraic numbers of
... |(L · L · Ld · . . . · L2 )| < C |x| it gives, that they are contained in finite number of hyperplanes. Now we will prove, that every hyperplane may contain only −d finite number of solutions even of the first inequality - |L(x)| < c1 |x| . Lemma 5.2.1. For every linear form L(x) = β0 x0 +. . .+βd x ...
... |(L · L · Ld · . . . · L2 )| < C |x| it gives, that they are contained in finite number of hyperplanes. Now we will prove, that every hyperplane may contain only −d finite number of solutions even of the first inequality - |L(x)| < c1 |x| . Lemma 5.2.1. For every linear form L(x) = β0 x0 +. . .+βd x ...
PDF9 - Pages
... For testing whether a number is prime, we pick a number which is relatively prime to and see if the equality holds. If it fails for any value of , then is composite. If the equality holds for many values of , then we can say that p is probably prime. The presence of Carmichael numbers prev ...
... For testing whether a number is prime, we pick a number which is relatively prime to and see if the equality holds. If it fails for any value of , then is composite. If the equality holds for many values of , then we can say that p is probably prime. The presence of Carmichael numbers prev ...
Overpseudoprimes, and Mersenne and Fermat Numbers as
... where p is prime. In this form numbers Mp , at the first time, were studied by Marin Mersenne (1588–1648) around 1644; see Guy [3, §A3] and a large bibliography there. In the next section, we introduce a new class of pseudoprimes and we prove that it just contains the odd numbers n such that |2|d is ...
... where p is prime. In this form numbers Mp , at the first time, were studied by Marin Mersenne (1588–1648) around 1644; see Guy [3, §A3] and a large bibliography there. In the next section, we introduce a new class of pseudoprimes and we prove that it just contains the odd numbers n such that |2|d is ...
A Pascal-Type Triangle Characterizing Twin Primes
... (3) and (8). Of course, these do look rather convoluted compared with the triangular recurrence for the Pascal triangle. Given the denominators on the right-hand sides of (10) and (11), it is by no means clear that the numbers a(k, s) and b(k, s), if they were defined in this way, should all be inte ...
... (3) and (8). Of course, these do look rather convoluted compared with the triangular recurrence for the Pascal triangle. Given the denominators on the right-hand sides of (10) and (11), it is by no means clear that the numbers a(k, s) and b(k, s), if they were defined in this way, should all be inte ...
Extra Examples — Page references correspond to locations of Extra
... At this point it is not clear how to proceed in order to conclude that n is even. (We could fashion such a proof, but it would require some additional knowledge about primes and divisibility.) It would be wise to look at a proof by contraposition. Suppose n is not even. Therefore n is odd. Therefore ...
... At this point it is not clear how to proceed in order to conclude that n is even. (We could fashion such a proof, but it would require some additional knowledge about primes and divisibility.) It would be wise to look at a proof by contraposition. Suppose n is not even. Therefore n is odd. Therefore ...
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"".