386: Finding Primes the Euler Way

Trying to find a prime number seems difficult. You have to test it against divisibility by all of the primes that are smaller. But what if there were a formula?

Take a counting number, multiply it by itself, subtract it from the result and then add 41

1x1 = 1; 1-1 = 0; 0+41 = 41. That's prime.
2x2 = 4; 4-2 = 2; 2+41 = 43. That's prime.
3*3 = 9; 9-3 = 6; 6+41 = 47. That's prime, too.
4*4=16; 16-4=12; 12+41= 53. That's also prime.

How good is this idea?
Will it ever fail?

350: Primes

This little puzzle, via @mathmovesu, asks for three consecutive prime numbers.

Is the guess and check method the best way to go here?

Which prime numbers are candidates and which ones can we safely ignore?

276: Primes' Product and Sum

The really difficult questions on the SAT (the ones at the end of each section) are often not really that difficult -- they just seem to be hard to answer because they're hard to read and understand because of the formal mathematical writing style and the complexity of the sentence. Often the best tactic is to plug in some numbers for the vague-sounding variables and make the problem more concrete.

"The positive number k is the product of four different positive prime numbers.
If the sum of these four prime numbers is a prime number greater than 20,
what is the least possible value for k?"

So what can we do with this one to make it simpler to understand the question?
Where would you start?