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.
The formula n? - n + 41 will give primes for n = 1, 2, 3
, 40 but not for n = 41. Which of the primes less
than 100 are given by this formula?
To find the primes less than 100 that are given by the formula n² - n + 41, we can simply substitute each value of n from 1 to 100 into the formula and check if the result is a prime number.
For n = 1:
1² - 1 + 41 = 41
41 is a prime number.
For n = 2:
2² - 2 + 41 = 43
43 is a prime number.
For n = 3:
3² - 3 + 41 = 47
47 is a prime number.
For n = 4:
4² - 4 + 41 = 53
53 is a prime number.
For n = 5:
5² - 5 + 41 = 61
61 is a prime number.
For n = 6:
6² - 6 + 41 = 71
71 is a prime number.
For n = 7:
7² - 7 + 41 = 83
83 is a prime number.
For n = 8:
8² - 8 + 41 = 97
97 is a prime number.
For n = 9:
9² - 9 + 41 = 113
113 is greater than 100, so we stop here.
Therefore, the prime numbers less than 100 that are given by the formula n² - n + 41 are 41, 43, 47, 53, 61, 71, 83, and 97.