SYSTEMATIC MATHEMATICS

Number Theory

This course contains 44 visible knowledge chapters in nine content-sized units. Why does division leave one unique remainder, why does that create a unique base expansion, how can an integer equation have no solution, and how can a huge power be reduced without writing the huge number? Begin with whole-number division and turn each calculation into a statement with an exact witness. Gcd, lcm, p-adic valuation, congruence, inverse, residue, Diophantine, primitive root, convolution, and descent are introduced from concrete equations before shorthand is used. The course proves infinitely many primes, classifies exactly which moduli have primitive roots, lifts simple roots to prime powers, proves both the prime and full-integer two-square theorems, certifies continued-fraction approximation, distinguishes Carmichael deception from strong compositeness witnesses, proves Wilson’s exact criterion, and proves sieve correctness. Its final modular example stays small enough to audit by hand and is never presented as real cryptographic security.

Before this course: Completed Proof, Logic & Set Theory, plus elementary integer arithmetic from Mathematical Foundations and finite counting from Discrete Mathematics. The course assumes direct, contrapositive, contradiction, induction, quantified statements, sets, functions, equivalence relations, and finite sums. Abstract Algebra Chapters 1–3 are helpful but not required: residue classes, inverses, multiplicative structure, and quotient language are reconstructed locally. No calculus, programming language, computer algebra system, cryptography course, or prior number theory is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Divisibility is an existential integer relation

Objective: Without looking at the worked lines, prove or verify this exact claim: “If a∣b and a∣c, then a∣(mb+nc) for all integers m,n.” Then solve “Show 7 divides 3·35−2·14 and give the quotient witness.” and check the result in the original equation, congruence, factorization, or algorithm.

Writing a∣b means a scales by some integer to equal b; the quotient witness matters, and zero cases must be handled explicitly.

Read the notation before using it: For integers a,b, a∣b means there exists k∈Z with b=ak. The statement to establish is: If a∣b and a∣c, then a∣(mb+nc) for all integers m,n. Every quotient, remainder, coefficient, exponent, residue, or factor that makes a claim true is kept as a checkable witness.

The worked question is: Show 7 divides 3·35−2·14 and give the quotient witness. Follow the numbered exact-arithmetic lines, then compare the result with the original statement. The failure boundary is shown separately so that a familiar formula is not used after one of its hypotheses has disappeared.

If a∣b and a∣c, then a∣(mb+nc) for all integers m,n.

Choose witnesses b=ar and c=as.

Substitute into mb+nc=a(mr+ns).

Since mr+ns is an integer, the definition gives divisibility.

Show 7 divides 3·35−2·14 and give the quotient witness.

  1. Compute 35=7·5 and 14=7·2.
  2. Then 3·35−2·14=7(15−4).
  3. The expression equals 7·11=77.

Result: Yes; the quotient witness is 11.