50

Would you prefer a game with a 90% chance of winning, or one with a 10% chance of losing?

The movie below shows the time evolution of a solution of the Cahn-Hilliard equation. Starting from an almost homogenous mixture of two alloys, the equation dynamics separate the mixture into its pure components (black and white). Later in time small regions containing the same alloy melt together and build larger and larger regions until the equation reaches a stationary state.

Step 3

Show that if N does not lie in our array, then 2N+1 is prime.

Got it, on to the conclusion;
I'm stuck, give me a hint;
Back to the main article.

Step 2

Show that if N lies in the array, then 2N + 1 is not prime.

Got it, on to the next step;
I'm stuck, give me a hint;
Back to the main article.

Step 1

Find a general formula describing the elements of the array.

Got it, on to the next step;
I'm stuck, give me a hint;
Back to the main article.

Step 3 — Solution

First turn the statement "if N does not lie in the array, then 2N + 1 is prime" into its logical equivalent "if 2N + 1 is not prime, then N does lies in the array". It's this second statement we'll prove.

Step 2 — Solution

Since N lies in the array, it is equal to (2n+1)m+n for some n and m. Use this to work out 2N+1 in terms of m and n:

2N+1=2[(2n+1)m+n]+1=2(2n+1)m+ 2n+1=(2n+1)(2m+1).

Step 1 — Solution

The first row starts with the number 4. The starting number of any other row is 3 steps on from that of the previous one. This tells us that the first number in row m is of the form

Hint for step 3

Turn the statement "if N does not lie in the array, then 2N + 1 is prime" into its logical equivalent "if 2N + 1 is not prime, then N lies in the array".

Got it, on to the conclusion;
No way, show me the solution.
Back to the main article

Hint for step 2

Numbers that are not prime can be factorised into two integers, both greater than 1.

Got it, on to the next step;
No way, show me the solution.
Back to the main article

Hint for step 1

What is the first entry of the mth row in terms of m? What is the difference between successive numbers in the mth row in terms of m? And how do you express the nth entry of a row in terms of n and m?

Got it, on to the next step;
No way, show me the solution.
Back to the main article