Prove that sum of 2 consecutive prime numbers both greater than 2, can be expressed as a product of 3 integer factors, all greater than 1
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Today we will prove an interesting property about consecutive prime numbers. We want to show that when we add two consecutive primes both greater than 2, the result can always be written as a product of exactly three integer factors, each greater than 1. The key insight is that all primes greater than 2 are odd numbers, so their sum is always even, giving us the factor 2 to start with.
Let's establish the mathematical foundation for our proof. We start with two consecutive prime numbers, both greater than 2. Since the only even prime is 2, all primes greater than 2 must be odd numbers. When we add two odd numbers together, the result is always even. This means our sum S can be written as 2 times some integer k, where k equals the average of our two consecutive primes.
Now let's analyze the possible values of k. The smallest case occurs when we have the consecutive primes 3 and 5, giving us k equals 4. In this case, the sum is 8, which factors as 2 times 2 times 2. For larger consecutive primes, k will be at least 6. The key insight is that when k is greater than 4, it must be composite. This happens because k is either one more than an odd prime, making it even and composite, or k lies strictly between two consecutive primes, which means it cannot be prime itself.
Now we complete our proof. Since we've established that k is composite whenever k is greater than 4, we can write k as a product of two integers b and c, both greater than 1. Substituting this back into our expression for S, we get S equals 2 times b times c. This gives us exactly what we wanted: the sum S expressed as a product of three integer factors - namely 2, b, and c - where all three factors are greater than 1. This completes our proof that the sum of any two consecutive primes greater than 2 can always be factored into exactly three factors, each greater than 1.
To summarize what we have proven: any sum of two consecutive prime numbers greater than 2 can be expressed as a product of exactly three integer factors, each greater than 1. The key insights were recognizing that such primes are odd, their sum is even, and the resulting quotient is composite in most cases, providing the necessary factorization structure.