Study Number Theory and Prime Numbers: 132 Flashcards in Two Levels

Deck UJ5237 "Number Theory & Prime Numbers" on Flip It holds 132 cards, and the thing to know before your first session is that it is two decks glued together. Cards 1-60 are a graduate-flavoured tour that ends in algebraic number fields and Faltings' theorem; cards 61-132 restart from "Is 1 a prime number?" and stay elementary. Study them back to back and the difficulty curve collapses halfway through. Split them and both halves work.

The upper half: cards 1-60

Terms to define, not questions: divisibility, the Euclidean division lemma, gcd and the Euclidean algorithm, linear Diophantine equations, the Fundamental Theorem of Arithmetic, Euclid's lemma, congruences and residue classes, Fermat's Little Theorem, Euler's totient φ(n), the Chinese Remainder Theorem, Pythagorean triples, perfect and amicable numbers, Wilson's theorem, the Riemann zeta function, quadratic reciprocity and the Legendre symbol, continued fractions, Pell's equation, algebraic integers, number fields, ideals, primitive roots, σ(n), τ(n), the Möbius function, Bertrand's postulate and Lagrange's four-square theorem.

The lower half: cards 61-132

Prime trivia and named conjectures, mostly in question form: trial division, prime gaps, Fermat and Euclid numbers, Dirichlet's theorem, Green-Tao, Sophie Germain and safe primes, Miller-Rabin and AKS, Carmichael numbers, Lucas-Lehmer, GIMPS, Chen's and Vinogradov's theorems, Brun's theorem, π(n) and the logarithmic integral, Gaussian primes, semiprimes, the Basel problem, Landau's problems, Mills' constant, and the Andrica, Cramér, Polignac, Legendre, Brocard and Oppermann conjectures.

Eight cards from the deck

Suspend the duplicates at the seam

The halves were written independently, so a dozen facts appear twice — once as an abstract statement, once as a chatty question. Fermat's Little Theorem (18 and 78), the Sieve of Eratosthenes (13 and 80), Goldbach (33 and 73), twin primes (34 and 72), coprimality (5 and 79), the Riemann hypothesis (29 and 84), the prime number theorem (27 and 75), Euclid's infinitude theorem (10, 66, 67), φ(p) (19 and 122). Two being the only even prime is asked three times (62, 71, 132), 1 not being prime twice (61, 64). Keep whichever version carries the formula or the proof sketch, suspend the other, or a tenth of your reviews will teach you nothing.

Rewrite two cards

Brocard's conjecture appears as "between any consecutive perfect squares n² and (n+1)², there are at least 4 primes". That is false as written — between 1 and 4 there are two — and it collides with the Legendre card two rows earlier. The real claim is about squares of consecutive primes: at least four primes between p_n² and p_n+1². Second, "Can every even number > 10 be written as a sum of primes?" is answered with a statement about six primes; put "as a sum of six primes" in the front, since as asked Goldbach already settles it.

More math decks: mathematical constants, squares, cubes and roots and the multiplication tables.

Deck code UJ5237 "Number Theory & Prime Numbers" on Flip It.