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...
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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 =...
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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) =...
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Tuesday, March 15, 2016
Quotation about teaching math.
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...
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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...
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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...
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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?
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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...
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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).
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Wednesday, October 28, 2015
Two Theorems:
Two Theorems: [1] 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. [2]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...
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