Proof Conjecture 11 Note that for t 12 S n the gcd 12 n 1 Let d gcd 12 n for

# Proof conjecture 11 note that for t 12 s n the gcd 12

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Proof. (Conjecture 1.1). Note that for t = 12 S + n , the gcd (12 , n ) > 1. Let d = gcd (12 , n ) for some integer d 2. Then t = d [(12 /d ) S + ( n/d )] where 12 /d and n/d must be integers since d is a divisor of both 12 and n . If d = 12, then t is divisible by every factor of 12. This proves that 12 S + n has an integer factor other than 1 and itself, i.e., 12 S + n is never prime. From this proof if follows that if gcd (12 , r ) = d > 1 for t = 12 S + r , then t is not prime. However, in Conjecture 1.3 we acknowledged that 2 and 3 8
correspond to a prime interval once. This is because 2 and 3 are prime numbers to begin with. Once we go beyond the first octave above the starting pitch, any t having a remainder of 2 or 3 upon division by 12 will not be prime since 2 and 3 are factors of both 12 and 2 or 3. The only claim left to prove is the first statement within Conjecture 1.4. In congruence notation, this conjecture states that there are infinitely many primes p of the form p 1 or 5 or 7 or 11 (mod 12). We were able to prove that numbers of the form t = 12 k + n are never prime, but how does one prove numbers of a special form are frequently prime? We will study a useful method for proving the existence of an infinitude of primes in the following section. This method will aid us in proving that special forms of numbers contain infinitely many prime numbers. Also, we will take a look at one example of prime numbers in a special form keeping in mind that our goal is to prove the first statement within Conjecture 1.4. 3 An Infinitude of Prime Numbers Before we begin any proofs or examples, we must understand an all important theorem that will be utilized within every following theorem. Theorem 3.1. Fundamental Theorem of Arithmetic . Every positive in- teger n 2 can be expressed as a product of prime numbers. Furthermore, this representation is unique, apart from the order in which the primes occur. Proof. Omitted. This proof can be found in any elementary number theory textbook. Let us take a look at some numerical examples of 3.1. Example 3.2. Consider the following prime factorizations: 9
30 = 2 · 3 · 5 15 = 3 · 5 64 = 2 6 41 = 41 There exists a very effective method for proving the existence of an infini- tude of primes. Around 300 BCE, Euclid proved the existence of an infinitude of primes by first assuming that there is a finite number of primes. Next, he constructed a number based on the product of all primes plus a remainder. Upon studying the prime factorization of this number, he arrived at a contra- diction within the proof. From this contradiction, the only conclusion is that there must be an infinitude of primes. This method motivates the proof of every theorem involving primes of a special form throughout Section 4. Let us begin our study of primes with Euclid’s famous theorem: Theorem 3.3. There is an infinite number of primes.

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