Pythagorean Triples Historical Context: Suggested Readings
... • The difference between successive squares of integers will be the odd numbers in sequence (e.g. 32 + 7 = 42, 42 + 9 = 52, 52 + 11 = 62) Furthermore, the equation n2 + (2n + 1) = (n+1)2 can be used to generate Pythagorean triples, but the term (2n+1) must be viewed as a perfect square. 1. Let m2 = ...
... • The difference between successive squares of integers will be the odd numbers in sequence (e.g. 32 + 7 = 42, 42 + 9 = 52, 52 + 11 = 62) Furthermore, the equation n2 + (2n + 1) = (n+1)2 can be used to generate Pythagorean triples, but the term (2n+1) must be viewed as a perfect square. 1. Let m2 = ...
Tools of Algebra (Review of Pre
... Variable- A variable is a _______________, usually a ______________, that represents one or more numbers. Algebraic Expressions- An algebraic expression is a ______________________ _________________ that can include _______________________, ________________________, and ______________________ ______ ...
... Variable- A variable is a _______________, usually a ______________, that represents one or more numbers. Algebraic Expressions- An algebraic expression is a ______________________ _________________ that can include _______________________, ________________________, and ______________________ ______ ...
online page proofs
... Each row of Pascal’s triangle contains the coefficients in the expansion of a power of (a + b). Such expansions are called binomial expansions because of the two terms a and b in the brackets. Row n contains the coefficients in the binomial expansion (a + b) n. To expand (a + b) 4 we would use the b ...
... Each row of Pascal’s triangle contains the coefficients in the expansion of a power of (a + b). Such expansions are called binomial expansions because of the two terms a and b in the brackets. Row n contains the coefficients in the binomial expansion (a + b) n. To expand (a + b) 4 we would use the b ...
mathematical problem solving
... This means that our conjecture that the sum of n consecutive numbers is a multiple of n is in general false, because it is false for n = 4, which is therefore a counterexample disproving our conjecture. Just to rub it in, and we should probably have done this first: the conjecture is not true for n ...
... This means that our conjecture that the sum of n consecutive numbers is a multiple of n is in general false, because it is false for n = 4, which is therefore a counterexample disproving our conjecture. Just to rub it in, and we should probably have done this first: the conjecture is not true for n ...
I. BASIC PERRON FROBENIUS THEORY AND INVERSE
... is well defined (vA 6= 0 because no row of A is zero) and continuous. By Brouwer’s Fixed Point Theorem, this map has a fixed point, which must be a nonnegative eigenvector of A for some positive eigenvalue, λ. Because a power of A is positive, the eigenvector must be positive. STEP 2: stochasticize ...
... is well defined (vA 6= 0 because no row of A is zero) and continuous. By Brouwer’s Fixed Point Theorem, this map has a fixed point, which must be a nonnegative eigenvector of A for some positive eigenvalue, λ. Because a power of A is positive, the eigenvector must be positive. STEP 2: stochasticize ...
Full text
... in the golden ratio base, so this alternating pattern can only repeat a finite number of times. Shifting, however, is problematic, since shifting continues an indefinite number of times until the two numbers involved are both positive. There is, however, an optimization that can be used to determine ...
... in the golden ratio base, so this alternating pattern can only repeat a finite number of times. Shifting, however, is problematic, since shifting continues an indefinite number of times until the two numbers involved are both positive. There is, however, an optimization that can be used to determine ...
Full text
... ing p r o c e s s , and (3) a change of entry. We note that no divisibility checks a r e used. One further comment. This process does not involve using any of the sequences A^ for m < (k-1) (k) k - 1 , but only the sequence A to produce A . As a result, those preceding sequences need not be saved. E ...
... ing p r o c e s s , and (3) a change of entry. We note that no divisibility checks a r e used. One further comment. This process does not involve using any of the sequences A^ for m < (k-1) (k) k - 1 , but only the sequence A to produce A . As a result, those preceding sequences need not be saved. E ...
series with non-zero central critical value
... an infinite number of the quadratic twists of an L-function associated to a cusp form will have nonzero central critical value. In [20], Ono has shown several examples of cusp forms f associated to elliptic curves such that for a positive density of the primes p, the pth quadratic twist of L(f, s) w ...
... an infinite number of the quadratic twists of an L-function associated to a cusp form will have nonzero central critical value. In [20], Ono has shown several examples of cusp forms f associated to elliptic curves such that for a positive density of the primes p, the pth quadratic twist of L(f, s) w ...
