Mathematics Introduction and Proofs

High School Mathematics: Focuses on solving equations and computing answers to numerical questions.
College Mathematics: Encompasses a broader range of topics including sets, functions, and other mathematical objects.
Commonality: Both levels rely on deductive reasoning to derive answers.

Definition: Deductive reasoning involves deriving conclusions based on premises or facts known to be true. In mathematics, this typically manifests as proofs.
Purpose of the Book: Aids in developing mathematical reasoning and proficiency in reading and writing proofs.
Structure: Detailed study of proofs begins later, but initial examples introduced to provide familiarity.

Subject of Proofs: Focused on prime numbers, defined as integers greater than 1 that cannot be expressed as a product of two smaller positive integers. For instance,
- Example of Non-Prime: 6 (because 6 = 2 × 3)
- Example of Prime: 7

Initial Findings: Exploring integers from 2 to 10 revealed a pattern regarding primes and the expression of the form 2n - 1.
Table Findings: If n is prime, 2n - 1 appears prime.

Conjecture 1: If n is prime and greater than 1, then 2n - 1 is prime.
Conjecture 2: If n is not prime and greater than 1, then 2n - 1 is not prime.

Counterexample: Check primes like 11, 23, and 29, and demonstrate that 2n - 1 yields non-prime results (e.g., for n = 11, 2^11 - 1 = 2047 = 23 × 89).
Conclusion: Conjecture 1 is incorrect as it's disproved by the existence of counterexamples.

Counterexample Search: No counterexamples found up to 30 for Conjecture 2, but failure to find one does not verify correctness.
Need for Proof: Until proven, we can’t ascertain truth.

Proof Strategy: Since n is not prime, n can be factored into positive integers a and b.
- Let x = 2b - 1 and a relation for y expressed in a summative form.
- $x imes y = 2n - 1$
Conclusion: Since n can be expressed this way, 2n - 1 cannot be prime if n isn’t prime.

For n = 12 (n is not prime):
- Calculate $x = 2b - 1 = 2 imes 6 - 1 = 15$
- Calculate $y$ .
- Verify $x imes y = 15 imes 273 = 4095$ .
- Confirming the structure of the proof.

Definition: Mersenne primes are primes of the form 2^n - 1.
Remark: Not all 2^n - 1 values are prime.
Current Knowledge: As of April 2005, largest known Mersenne prime is 2^25,964,951 - 1 (7,816,230 digits).

Definition: A perfect number equals the sum of its proper divisors (e.g., 6, 28).
Connection with Mersenne Primes: Euclid proved that if 2^n - 1 is prime, then 2^(n-1)(2^n - 1) is perfect.
Euler’s Contribution: He established that every even perfect number arises in this way, linking perfect numbers to Mersenne primes.

Proof of Euclid: To show infinitude of primes:
- Assume finitely many primes p1, p2,…, pn. Define $m = p1 * p2 * … * pn + 1$ .
- Illustrating m is either prime or has a prime divisor not in the initial list, contrary to assumptions about finiteness.

Theorem: For every positive integer n, there exists a stretch of n consecutive positive integers containing no primes.
Proof Outline: Define $x = (n+1)! + 2$ and demonstrate that none of the integers in $[x, x+n-1]$ are prime.

At any integer $x+i$ :
- The construction leads to expressions of integers showing non-primality.

Definition: Pairs of primes that differ by 2 (e.g., (5,7)).
Investigation: Whether there are infinitely many such pairs remains an open question.

Conjecture 1: If n is prime and greater than 1, then 2n - 1 is prime.
Conjecture 2: If n is not prime and greater than 1, then 2n - 1 is not prime.

To test Conjecture 1, we need to check specific prime numbers. Here's the step-by-step way to do this:

Select Prime Numbers: Start with primes such as 11, 23, and 29.
Calculate 2n - 1: For each selected prime, compute the expression 2n - 1.
- For n = 11, this yields:
  - $2^{11} - 1 = 2047$ which can be factored as $23 imes 89$ (not prime).
- For n = 23, we find:
  - $2^{23} - 1 = 8388607$ and can check if this number is prime.
- For n = 29, we compute:
  - $2^{29} - 1 = 536870911$ and evaluate its primality as well.
Conclusion: Since we found at least one counterexample (2047), we can conclude that Conjecture 1 is incorrect because there are primes n that result in 2n - 1 being non-prime.

To assess Conjecture 2, we follow a similar approach but focus on composite numbers.

Select Non-Prime Numbers: Choose some integers greater than 1 that are known to be non-prime, like 4, 6, 8, etc.
Calculate 2n - 1: For each non-prime n, compute 2n - 1.
- For n = 4, we calculate:
  - $2^{4} - 1 = 15$ (which equals $3 imes 5$ and is non-prime).
- For n = 6, we find:
  - $2^{6} - 1 = 63$ (which equals $7 imes 9$ and is also non-prime).
Check Further Values: Continue checking for other values such as 9, 10, and 12, all yielding non-prime results.
Conclusion: We cannot find any counterexamples up to 30, suggesting Conjecture 2 may be true, but without conclusive proof, we remain uncertain about its validity at larger numbers.

Understanding Non-Primes: If n is not prime, it means n can be expressed as a product of two integers, say a and b, where a is greater than 1 and b is greater than 1.
Expression Setup: We let $x = 2b - 1$ .
Linking x and n: Given $2n - 1 = 2(ab) - 1 = 2a imes b - 1$ makes it clear that if n is not prime, the resultant can't be prime either since we can find divisors.
Final Check: To prove conclusively, substitute specific values of a and b proving that the result remains