How much would we have to fortify the Lonely Runner problem in order to find a counter-example speed assignment, by adding extra runners, while keeping the loneliness criteria the same? For N Runners,

I use my base 2 factoring algorithm to confirm that 31 is prime. z = 31 z = (1 + 2 + 4 + 8 + 16) = (1 + 2 * x1 + 4 * x2) * (1 + 2 * y1 + 4 * y2 + 8 * y3 + 16 * y4) 1 + 2*(x1 + y1) + 4*(x2+x1*y1+y2) +

Today I thought of this alternative base 2 analysis of factoring. The number of steps to find the divisors should be a function of the logarithm of the number for which we wish to find the divisors, b

Hi All Here is a conjecture about primes. All primes p from 2 to 113 can be written as p=x*(x+k)+c or p=x*(x+k)-c where k is either 0 or 1 and c is either 1,2,3. The prime 127 is the first exception t

Using the integers 0 through 3, you can make a table of addition. 0 1 2 3 This corresponds to (2^4 - 1) = (2^0 + 2^1) * (2^0 + 2^2) You cannot make a corresponding addition table using integers 0 thro

Hi All I have an idea for a puzzle. The idea is to place the numbers from 1 to N^2 in a N*N square and note the general distance from k to k+1 in terms of x coordinates and y coordinates. The first nu

The following algebraic identity shows a deeper relationship between addition and multiplication. (q1+t2*t1*q2)*((t3 * t2+1) * q2) = (q1*q2) + (t2*q2) * (t3*t2*t1*q2 + t3*q1 + t1*q2) It is a formula f

-----Original Message----- From: "kermit@..." <kermit@...> Sent: Saturday, November 12, 2022 6:40pm To: yahoo.com@groups.io Subject: FW: RE: [UnsolvedProblems] Puzzle Kim, what would be the score for

Hello Kim. If I understand your puzzle correctly, there is probably a trivial solution for any order square. For the 4 by 4 square, I submit 7 8 9 10 6 1 2 11 5 4 3 12 16 15 14 13 Kermit -----Original

Hello Daniel. You are correct. An example of your point is (a * x^2 - b*x + c) * (a * x^2 + b*x + c) = (a* x^2 + c)^2 - b^2 * c*2 = a^2 * x^4 + (2 * a * c - b^2) * x^2 + c^2 a*x^2 - b*x + c = 1 a*x^2

3 = 2+1 implies that x+1 is irreducible. 5 = 2^2 + 1 implies that x^2+1 is irreducible. 7 = 2^2+2+1 implies that x^2 + x + 1 is irreducible. 11 = 2^3 + 2 + 1 implies that x^3 + x + 1 is irreducible. 1

Hi all Can numbers of this form be parametrized ? where x,z,r and Q are integers and x is odd. More specifically I want to know if there are infinetely many odd integers W which is not equal to any nu

Introducing the ordered lists: How is an ordered list, [0, 1, 2, 3], made into a puzzle? The addition operation is defined on the ordered lists. Addition Table [0,1] [0,2] [0,1,2] [0,3] [0,1] [0,3] [0

Hi All, I'm back. Finally a more simple and clear proof for Fermat the last (I hope) was found with my CMA. From Sum to Step Sum, then to the limit where only the integral can arrive from n=3. Here th

The ring of (lists of numbers). Define a (list of numbers) to be a list of numbers, [a1, a2, a3, ...] An equivalent term for (list of numbers) is number list. Define the sum of two number lists, [a1,

Re factoring algorithms: a good test is that they have to be able to easily factor 20+ digit numbers, since the simplest sieve methods can handle these in less than a millisecond. Tim

How can I apply the concept of list number factorization to factor large numbers? I thought I should study examples where I knew the factors. Perhaps some principles of deriving the factors would emer

Factoring by Addition Table Transform lists by replacing an integer by 2 copies of the next smaller integer. Equate integer to list via exponents in base 2 representation. 9 = [0,3] = [0,2,2] = [0,1,1

Hello friends. The attached Excel file applies the mod 24 factoring algorithm that I developed today. If z is prime, or if the smallest divisor > 1 is less than 24, p = 0 finds the factors. If z is co

Hi All Suppose we have 5 different elements which we will put in to a ordered list of 3 elements. In how many ways can this be done ? If repetitions are allowed then the answer is 5*5*5=5^3=125 Now wh