# Mathematics

### Thursday, August 11, 2016

#### Number of primes between consecutive squares

Friend David Broadhurst corrects me as follows: Kermit suggested: > The approximate number of primes between m^2 and (m+1)^2 is > (2m+1) * product over all positive primes, p, less than or equal > to m of (p-1)/p No. That is Chebyschev's conundrum. As m tends to infinity, the left hand side is... Sign in to see full entry.

### Friday, July 29, 2016

#### If x+y = x y = 3, then what is x^3 + y^3?

If x+y = x y = 3, then what is x^3 + y^3? Solution 1: 27 = (x+y)^3 = x^3 + 3 x^3 y + 3 x y^2 + y^3 = (x^3 + y^3) + 3 x y (x+y) = (x^3 + y^3) + 3 * 3 * 3 = (x^3 + y^3) + 27 Therefore x^3 + y^3 = 0 Solution 2: x+y = x y = 3 (t-x)(t-y) = t^2 - (x+y) t + x y The quadratic equation, t^2 - (x+y) t + x y =... Sign in to see full entry.

### Wednesday, April 20, 2016

#### Examples of Sum of products equal to sum of products

For all integers A, B, C, D, the sum of products, (A-B)*(C+D) + (B-D)*(A+C) is equal to the sum of products (A+B)*(C-D) + (A-C)*(B+D) Example 1: Let A = 11, B = 7, C = 5, D = 3 (A-B)*(C+D) + (B-D)*(A+C) = (11-7)*(5+3) + (7-3)*(11+5) = 4*8 + 4*16 = 32 + 64 = 96 (A+B)*(C-D) + (A-C)*(B+D) =... Sign in to see full entry.

### Tuesday, March 15, 2016

Quotation about teaching math. If teachers would only encourage guessing. I remember so many of my math teachers telling me that if you guess, it shows that you don’t know. But in fact there is no way to really proceed in mathematics without guessing. You have to guess! You have to have intuitive... Sign in to see full entry.

### Sunday, March 13, 2016

#### How to make the sum or difference of two integers to be prime

3 + 2 = 5 9 - 4 = 5 3 + 4 = 7 9 - 2 = 7 3 + 8 = 11 27 - 16 = 11 9 + 4 = 13 27 - 14 = 13 9 + 8 = 17 27 - 10 = 17 9 + 10 = 19 27 - 8 = 19 9 + 14 = 23 27 - 4 = 23 27 + 2 = 29 30 - 1 = 29 30 + 1 = 31 30 + 7 = 37 30 + 11 = 41 30 + 13 = 43 30 + 17 = 47 The above representations of prime integers... Sign in to see full entry.

### Tuesday, March 1, 2016

#### Interesting math puzzle

I made up an interesting math problem today. The product of x and (x-10) is equal to the product of y and (y+1). There is one integer solution. Can you discover it? What is it that makes this puzzle particulary interesting? It is that x and y that satisfy this equality also must satisfy that the... Sign in to see full entry.

### Tuesday, December 1, 2015

#### Puzzle for December 1, 2015

Timothy spent all his money in five stores. In each store, he spent \$1 more than half of what he had when he came in. How much did Timothy have when he entered the first store? Sign in to see full entry.

### Wednesday, November 11, 2015

#### Calendar dates

Calendar dates The 7th day of the 3rd month, March, is the same day of the week as the 7th day of the 11th month, November, is the same day of the week as the 11th day of the 7th month, July, is the same day of the week as the 5th day of the 9th month, September, is the same day of the week as the... Sign in to see full entry.

### Friday, October 30, 2015

#### Sum of two products is product

Hello dear Kabu. Thank you for being among my best students. Here is another puzzle I made recently. Show why, for all numbers, A, B, C, that (A-B)(A+B) + (B-C)(B+C) = (A-C)(A+C). Sign in to see full entry.

### Wednesday, October 28, 2015

#### Two Theorems:

Two Theorems:  Prove that if M is any integer greater than 2, then there exist integers n1 and n2 such that each of n1 and n2 is the sum of two square integers, and M =n1 - n2. Prove that for all numbers a1, a2, b1, b2, (a1 a2 - b1 b2)^2 + (a1 b2 + b1 a2)^2 + ((a1^2 + b1^2 - a2^2 - b2^2)/2)^2... Sign in to see full entry.

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