It is likely that no one will ever find a formula that will give all the primes less than an arbitrary number. The formulas in exercises 34 and 35 produce primes for a while, but eventually they produce composite numbers.
34. For which of the whole numbers n = 2 to 7 is 2" - 1
a prime?
To determine which values of n from 2 to 7 yield a prime number when plugged into the formula 2^n - 1, we can calculate the results for each value of n:
For n = 2:
2^2 - 1 = 4 - 1 = 3 (prime)
For n = 3:
2^3 - 1 = 8 - 1 = 7 (prime)
For n = 4:
2^4 - 1 = 16 - 1 = 15 (composite)
For n = 5:
2^5 - 1 = 32 - 1 = 31 (prime)
For n = 6:
2^6 - 1 = 64 - 1 = 63 (composite)
For n = 7:
2^7 - 1 = 128 - 1 = 127 (prime)
Therefore, the values of n that yield prime numbers when plugged into the formula are 2, 3, 5, and 7.