#### Number of primes between consecutive squares

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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.

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.

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.

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.

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.

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.

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.

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.

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.

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... **Sign in** to see full entry.