December 3, 2018

If n is the sum of three consecutive primes and is also the product of two 2-digit primes, what is the least possible value of n?

Let’s try the two 2-digit primes of least value, 11 and 13. These primes have a product of 143, and 143/3 is around 47. Two primes close in value to 47 are 43 and 53. So we try the sum 43 + 47 + 53 and see that it does, in fact, equal 143.

If p is the sum of three consecutive primes and also is the square of a prime, what is the least possible value of p

Since three consecutive primes can’t have an even sum, we know p is not 4. So let’s take a look at the squares of the primes beginning with the prime number three.
       32 = 9 and 9/3 is 3 but 2 + 3 + 5 ≠ 9
       52 = 25 and 25/3 is close to 7 but 5 + 7 + 11 ≠ 25
       72 = 49 and 49/3 is close to 17 and 13 + 17 + 19 does =
49.

Three consecutive primes, with values less than 200, have a sum equal to the product of two other primes that have a difference of 34. What is the value of the greatest of the three consecutive primes?

Let’s first find a pair of primes that differ by 34.
3 + 34 = 37, another prime. 3 × 37 = 111 and 111/3 is 37 but 31 + 37 + 41 ≠ 111
5 + 34 = 39, not prime.
7 + 34 = 41, another prime. 7 × 41 = 287 and 287/3 is close to 97 and 89 + 97 + 101 = 287.
The greatest of the three consecutive primes is
101.

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Page 2 contains ONLY PROBLEMS. ♦