![[Chap. 2] Pythagorean Triples (b) The table suggests that in every](http://s1.studyres.com/store/data/016288292_1-0734630555f52d6c6864c2800221c690-300x300.png)
[Chap. 2] Pythagorean Triples (b) The table suggests that in every
... This gives a primitive Pythagorean triple with the right value of b provided that B > 2r−1 . On the other hand, if B < 2r−1 , then we can just take a = 22r−2 − B 2 instead. (c) This part is quite difficult to prove, and it’s not even that easy to make the correct conjecture. It turns out that an odd ...
... This gives a primitive Pythagorean triple with the right value of b provided that B > 2r−1 . On the other hand, if B < 2r−1 , then we can just take a = 22r−2 − B 2 instead. (c) This part is quite difficult to prove, and it’s not even that easy to make the correct conjecture. It turns out that an odd ...
Mar - Canadian Mathematical Society
... consecutive integers that become four-digit n-gonal numbers, n > 3 and r > 2. For a greater challenge one can determine four, five, • • • digit pairs of consecutive integers that become n-gonal numbers non-trivially. Directions for further investigation include extensions of the basic problem we hav ...
... consecutive integers that become four-digit n-gonal numbers, n > 3 and r > 2. For a greater challenge one can determine four, five, • • • digit pairs of consecutive integers that become n-gonal numbers non-trivially. Directions for further investigation include extensions of the basic problem we hav ...
Induction
... = (n + 1)n+1 Hence, by the induction principle, ∀n ∈ N, if n > 1, then n! < n n . ♠ In the middle of the last century, a colloquial expression in common use was ”that is a horse of a different color”, referring to something that is quite different from normal or common expectation. The famous mathem ...
... = (n + 1)n+1 Hence, by the induction principle, ∀n ∈ N, if n > 1, then n! < n n . ♠ In the middle of the last century, a colloquial expression in common use was ”that is a horse of a different color”, referring to something that is quite different from normal or common expectation. The famous mathem ...
Full text
... E-mail address: [email protected] Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 E-mail address: [email protected] Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 E-mail address: [email protected] Department of Mathematics and Statistics ...
... E-mail address: [email protected] Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 E-mail address: [email protected] Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 E-mail address: [email protected] Department of Mathematics and Statistics ...
Could Euler have conjectured the prime number theorem?
... P∞life,1 culminating in such gems as his solution to the Basel problem of evaluating the sum n=1 n2 [Eul40], E41, and the pentagonal number theorem [Eul83], E541. Or perhaps he simply had no reason to believe that there would be any large-scale patterns in the primes. But my suspicion is that someth ...
... P∞life,1 culminating in such gems as his solution to the Basel problem of evaluating the sum n=1 n2 [Eul40], E41, and the pentagonal number theorem [Eul83], E541. Or perhaps he simply had no reason to believe that there would be any large-scale patterns in the primes. But my suspicion is that someth ...
Problem Solving: Consecutive Integers
... Objective To write equations to represent relationships among integers. ...
... Objective To write equations to represent relationships among integers. ...
THE FRACTIONAL PARTS OF THE BERNOULLI NUMBERS BY
... B 2k + Y 1/p is an integer, where the sum is taken over all primes p for which (p - 1) 2k . Several years ago one of us computed {B20) for 2 < 2k < 10000 and noted two curious irregularities in their distribution : (1) There were large gaps, e .g ., the interval [0 .167, 0 .315], which contained non ...
... B 2k + Y 1/p is an integer, where the sum is taken over all primes p for which (p - 1) 2k . Several years ago one of us computed {B20) for 2 < 2k < 10000 and noted two curious irregularities in their distribution : (1) There were large gaps, e .g ., the interval [0 .167, 0 .315], which contained non ...
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"".