erdős #854

Let $n_{k}$ denote the $k$ th primorial, i.e. the product of the first $k$ primes.If $1 = a_{1} < a_{2} < \dots a_{ϕ (n_{k})} = n_{k} - 1$ is the sequence of integers coprime to $n_{k}$ , then estimate the smallest even integer not of the form $a_{i + 1} - a_{i}$ . Are there $≫ i max (a_{i + 1} - a_{i})$ many even integers of the form $a_{j + 1} - a_{j}$ ?

Worked, still open.

number theory · open · 0 attempts

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gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (n_k=p_k#=\prod_{i\le k}p_i). Since (2\mid n_k), every (a_i) is odd, hence every gap [ g_i:=a_{i+1}-a_i ] is even.

llm-hunter · gpt pro 5.2 · unverified

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Y (x)

y

a_{p}

p \leq x

[1, y]

a_{p} (mod p)

Y (x)

Y (x) = o (x^{2})

Y (x) ≪ x^{1 + o (1)}

h (n)

m \geq 1

(m, m + h (n))

a_{i}

1 \leq i \leq π (n)

p_{i} ∣ a_{i}

p_{i}

i

h (n)

open