Is there an infinite set $A \subset N$ such that for every $a \in A$ there is an integer $n$ such that $ϕ (n) = a$ , and yet if $n_{a}$ is the smallest such integer then $n_{a} / a \to \infty$ as $a \to \infty$ ?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

machinery: euler-totient-preimage,totient-value-distribution,carmichael-totient-conjecture,prime-distribution,number-theory

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

If you write [ g(a)\ :=\ \min\\{n:\ \varphi(n)=a\\}\qquad(a\ \text{a totient}), ] then your question is asking whether there are infinitely many totients $a$ with [ \frac{g(a)}{a}\ \to\ \infty\quad\text{along an infinite set}. ] Equivalently, it asks whether for **every** constant (C>1) there exists a totient $m$ such …

candidate solution ↗

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_51 : answer(sorry) ↔ ∃ A : Set ℕ, ∃ n : A → ℕ,
      A.Infinite ∧
      (∀ a : A, IsLeast (φ ⁻¹' {(a : ℕ)}) (n a)) ∧
      Tendsto (fun a : A => (n a : ℝ) / (a : ℝ)) atTop atTop

formal-conjectures/51.lean ↗

oeis

A002202 — Values taken by totient function phi(m) (A000010).1,2,4,6,8,10,12,16,18,20,22,24,28,30,32,36,40,42,44,46,48,52,54,56,58,60,64,66,70,72,78,80,82,84,88,92,96,100,102,104,10 A014197 — Number of numbers m with Euler phi(m) = n.2,3,0,4,0,4,0,5,0,2,0,6,0,0,0,6,0,4,0,5,0,2,0,10,0,0,0,2,0,2,0,7,0,0,0,8,0,0,0,9,0,4,0,3,0,2,0,11,0,0,0,2,0,2,0,3,0,2,0,

links

#821Let

g (n)

count the number of

m

such that

ϕ (m) = n

. Is it true that, for every

ϵ > 0

, there exist infinitely many

n

such that

g (n) > n^{1 - ϵ} ?

A014197

status