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#36 Prime generators

set by Peter Rowlett

In 1772, mathematician Leonhard Euler noticed that the value of the polynomial x² + x + 41 is a prime number if you set x=0. He also spotted it is prime if x=1 and it keeps on giving primes for values of x up to 39.

If we use x² + x + 3 instead, and put x=0, we get 3. Using x=1 gives 5, but the pattern then breaks because x=2 gives 9, which isn鈥檛 prime. So, x² + x + 3 generates a sequence of two primes.

How many primes in a sequence are created by x² + x + 5 using x=0, 1, 2 and so on?

For a prime p, we can generate a sequence of primes using x² + x + p with x=0, 1, 2 and so on. Which value of p between 220 and 230 generates the longest sequence of primes?

I noticed that for all the values of p with a polynomial that generates multiple primes, p+2 is also prime (we call this a pair of twin primes). Why is this? Will it always be the case?

Solution next week

#35 Cube cuts

Solution

To make four pieces, we just need two cuts. One would cut a 3-by-3 slice off one side of the cube, then we just keep the pieces together and cut along any plane that passes through both.

If each cut you make splits the cubes in half, it should be possible to create as many as 2⁵ = 32 smaller pieces in five cuts. In this case, however, the first cut will always split the 27 cubes into pieces of size 9 and 18, and then there is no way to split the 18 cubes in only four cuts (even if we split them in half each time, 2⁴ is only 16).

If we can only cut one piece at a time, this will take 26 cuts: each cut increases the total number of pieces by 1.