Mathematics

Monday, May 18, 2015

Intermediate results for magic square of squares

Intermediate results for magic square of squares If A^2 = (2/3) G^2 + (2/3) H^2 -(1/3) J^2 B^2 = (2/3) G^2 - (1/3) H^2 + (2/3) J^2 C^2 = -(1/3) G^2 + (2/3) H^2 + (2/3) J^2 D^2 = -(2/3) G^2 + (1/3) H^2 + (4/3) J^2 E^2 = (1/3) G^2 + (1/3) H^2 + (1/3) J^2 F^2 = (4/3) G^2 + (1/3) H^2 -(2/3) J^2 A^2 B^2... Sign in to see full entry.

Wednesday, May 13, 2015

Integral( sin(x) exp(x) dx)

Integral( sin(x) exp(x) dx) = (1/2) (sin(x) - cos(x)) exp(x) Proof: Let y = (1/2) (sin(x) - cos(x)) exp(x) then y' = (1/2) (sin(x) - cos(x) exp(x) + (1/2) (cos(x) + sin(x) exp(x) = (1/2) (2 sin(x) ) exp(x) = sin(x) exp(x). Sign in to see full entry.

Wednesday, May 6, 2015

n to the fourth power plus 4

Can you factor n^4 + 4 If n = 1, then n^4 +4 = 1 + 4 = 5 = 1 * 5 if n = 2, then n^4 + 4 = 16 + 4 = 20 = 2 * 10 if n = 3, then n^4 + 4 = 81 + 4 = 5 * 17 If n^4 + 4 factors into two quadratic polynomials, p1 and p2, then we could require that p1(1) = 1 p1(2) = 2 p1(3) = 5 p2(1) = 5 p2(2) = 10 p2(3) =... Sign in to see full entry.

Tuesday, May 5, 2015

Factoring a few special 4th degree polynomials

N^4 + N^2 + 1 = (N^2 - N + 1)(N^2 + N + 1) N^2 means N squared. N^4 means N reaised to the fourth power. N^4 + 2 N^2 + 9 = (N^2 - 2 N + 3) (N^2 + 2 N + 3) N^4 + 3 N^2 + 4 = (N^2 - N + 2) (N^2 + N + 2) N^4 + 4 N^2 + 4 = (N^2 + 2)^2 N^4 + 5 N^2 + 9 = (N^2 - N + 3) (N^2 + N + 3) N^4 + 6 N^2 + 9 = (N^2... Sign in to see full entry.

Monday, May 4, 2015

factoring of special numbers.

4 + the fourth power of an odd positive integer greater than 1, is always composite. Let b be an odd positive integer greater than 1. Note that if b > 1, then (b^2 - 2 b + 2) is greater than 1. (b^2 - 2 b + 2) * (b^2 + 2 b + 2) = (b^2 + 2 - 2 b) * (b^2 + 2 + 2 b) = (b^2 + 2)^2 - (2 b)^2 = (b^4 + 4... Sign in to see full entry.

Saturday, April 18, 2015

How to make the sum of three squares make a square

The product of two number, each of which is the sum of two squares, will also be the sum of two squares. (a^2 + b^2)(c^2 + d^2) = (a^2 c^2 - b^2 d^2)^2 + (a^2 d ^2 + b^2 c^2)^2. It is also the difference of two squares. (a^2 + b^2)(c^2 + d^2) = ((a^2 + b^2 + c^2 + d^2)/2)^2 - ((a^2 + b^2 - c^2 -... Sign in to see full entry.

Thursday, April 2, 2015

Math opportunity test

The teacher called it an opportunity test! Diane half smiled as she tilted her head back and using her fingers, brushed her long black hair over and behind her ears. Mr Smiley called it an opportunity test because it gave everyone a chance to make an A for the year. It was Mr. Smiley's tradition.... Sign in to see full entry.

Saturday, March 28, 2015

Super Magic Squares

The following is a general formula for constructing a Supermagic 5 by 5 square. What makes it Supermagic? If you move the top row down to become the bottom row, it is still a magic square. Repeat this move 4 more times. Each time, the magic remains. If you move the left column to become the... Sign in to see full entry.

Monday, March 23, 2015

Changing a dollar

Occasionally one sees posted the claim that there are 293 ways to change a dollar. However, apparently no one has questioned this, even though it is simple to confirm for yourself whether it is true. I made an Excel spreadsheet to help me count accurately. I noted that in fact there are 292 ways to... Sign in to see full entry.

Monday, March 16, 2015

Linear transformation of Pythagorean Triple into Pythagorean Triple.

Linear transformation of Pythagorean Triple into Pythagorean Triple. If x^2 + y^2 = z^2, and q1 = (1/2)(a^2 - b^2 - c^2 + d^2), q2 = (a c - b d), q3 = (1/2)(a^2 - b^2 + c^2 - d^2), q4 = (a b - c d), q5 = (a d + b c), q6 = (a b + c d), q7 = (1/2)(a^2 + b^2 - c^2 - d^2), q8 = (a c + b d), q9 =... Sign in to see full entry.

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