Two of the many conjectures involving primes are given in exercises 36 and 37.
36. The mathematician Christian Goldbach (1690-1764)
conjectured that every odd number greater than 5 is the sum of three primes. Verify this conjecture for the following numbers: 21, 27, 31.
To verify Goldbach's conjecture for a given number, we need to check if the number can be expressed as the sum of three prime numbers.
Let's start with the number 21. We need to find three prime numbers whose sum equals 21. Possible combinations are:
2 + 2 + 17 = 21
5 + 5 + 11 = 21
7 + 7 + 7 = 21
In all three cases, we have three prime numbers whose sum equals 21. Therefore, Goldbach's conjecture is verified for 21.
Now let's move on to the number 27. Possible combinations are:
2 + 2 + 23 = 27
2 + 13 + 12 = 27
3 + 3 + 21 = 27
5 + 5 + 17 = 27
7 + 7 + 13 = 27
7 + 11 + 9 = 27
In all six cases, we have three prime numbers whose sum equals 27. Therefore, Goldbach's conjecture is verified for 27.
Finally, let's consider the number 31. Possible combinations are:
2 + 2 + 27 = 31
2 + 5 + 24 = 31
2 + 7 + 22 = 31
2 + 11 + 18 = 31
3 + 3 + 25 = 31
3 + 5 + 23 = 31
3 + 11 + 17 = 31
5 + 5 + 21 = 31
5 + 7 + 19 = 31
7 + 7 + 17 = 31
11 + 11 + 9 = 31
In all eleven cases, we have three prime numbers whose sum equals 31. Therefore, Goldbach's conjecture is verified for 31.
In conclusion, Goldbach's conjecture is verified for the numbers 21, 27, and 31.