Practice: Reading the mathematical literature

Reading the mathematical literature

Warm-upHuman

A preprint posted yesterday on the arXiv claims a proof of the twin prime conjecture. You are writing a survey. What is the appropriate way to mention it?

Choose one

Cite it as a preprint that claims a proof, and state that the conjecture remains open until the claim has been checked.State that the twin prime conjecture has been proved, citing the preprint.Ignore it, because only peer-reviewed papers count.Wait until it appears in a prestigious journal and then treat it as certainly correct.

Reading the mathematical literature

Warm-upHuman

You want reviews of the published papers on a topic, classified by subject, with their references and the later papers that cite them. Which resources are designed for this? (Choose all that apply.)

Choose all that apply

zbMATH OpenMathSciNetThe arXivThe OEIS

Reading the mathematical literature

Warm-upHuman

For $p = 29$ , Zagier's involution has the unique fixed point $(1, 1, z)$ . What is $z$ ?

Reading the mathematical literature

CoreComputational

List the set $S = \{(x, y, z) \in \N^3 : x^2 + 4yz = 17\}$ (positive integers). How many elements does it have?

Reading the mathematical literature

CoreHuman

Run Zagier's argument for $n = 9$ instead of a prime. Here $S = \{(1, 1, 2), (1, 2, 1)\}$ . Which step of the reconstructed proof breaks?

Choose one

Lemma 1:

(1, 2, 1)

satisfies

x = y - z

, a boundary case where Zagier's map is not defined, so the map is not an involution of

S

.

S

is infinite.The parity lemma fails.There is no candidate fixed point.

Reading the mathematical literature

CoreHuman

Which of these sentences from the introduction of a (hypothetical) paper assert results that the paper itself proves? (Choose all that apply.)

Choose all that apply

"We prove that every sufficiently large integer of the form … is the sum of a prime and a square-free number (Theorem 1.1).""By a theorem of Siegel [12], the class number grows like …""Assuming the generalised Riemann hypothesis, we show that the bound holds for all

n \ge 2

(Theorem 1.3).""Numerical computations up to

10^{12}

suggest that the constant

2

is optimal.""It is well known that the exceptional set is finite."

Reading the mathematical literature

CoreFormal

Which Lean statement says exactly what Zagier's note proves?

Choose one

∀ p : ℕ, p.Prime → p % 4 = 1 → ∃ a b : ℕ, a ^ 2 + b ^ 2 = p∀ p : ℕ, p % 4 = 1 → ∃ a b : ℕ, a ^ 2 + b ^ 2 = p∀ p : ℕ, p.Prime → ∃ a b : ℕ, a ^ 2 + b ^ 2 = p∃ a b : ℕ, ∀ p : ℕ, p.Prime → p % 4 = 1 → a ^ 2 + b ^ 2 = p

Reading the mathematical literature

CoreHuman

A paper states: " $|M(n)| \lt \sqrt n$ has been verified for all $1 \lt n \le 10^{16}$ ", where $M$ is the Mertens function. What may a careful reader conclude?

Choose one

That there is no counterexample to the Mertens conjecture up to

10^{16}

— and nothing beyond. In fact the conjecture is false (Odlyzko and te Riele, 1985).That the Mertens conjecture is true, since

10^{16}

cases cannot all be coincidences.That any counterexample must be larger than

10^{16}

, so the conjecture is "true in practice".Nothing, because computations are never reliable.

Reading the mathematical literature

StretchComputational

Turn Zagier's proof into an algorithm. Write two_squares(p) that, for a prime $p \equiv 1 \pmod 4$ , builds $S = \{(x, y, z) : x^2 + 4yz = p\}$ (positive integers), finds a fixed point $(x, y, y)$ of the swap $(x, y, z) \mapsto (x, z, y)$ , and returns the pair (x, 2*y).

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