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I am running a program to search for solutions of $$\varphi(pm+1)=\varphi(pm+p+1).$$

So far, for $m=1,\ldots,327$ solutions have been found (some relatively large).

(in the body of the question, $p$ is a natural number, not necessarily prime)

I would like to conjecture that for every $m \in \mathbb N$ there exists natural number $p \in \mathbb N$ such that $\varphi(pm+1)=\varphi(pm+p+1)$

• Is this known to be true (or easily deducible)?

• If not known to be true, is there some evidence that totient could have (or not have) this interesting property?

Update: I stopped the program at $m=407$ since it was taking a long long time to find solution for $m=407$ (if there is any), but all $m=1,\ldots,406$ have a solution.

Update 2: Currently, the program is running the code for approx. half an hour just for $m=490$, and yes, the solutions exist for $m=1,\ldots,489$

I am running a program to search for solutions of $$\varphi(pm+1)=\varphi(pm+p+1).$$

So far, for $m=1,\ldots,327$ solutions have been found (some relatively large).

(in the body of the question, $p$ is a natural number, not necessarily prime)

I would like to conjecture that for every $m \in \mathbb N$ there exists natural number $p \in \mathbb N$ such that $\varphi(pm+1)=\varphi(pm+p+1)$

• Is this known to be true?

• If not known to be true, is there some evidence that totient could have (or not have) this interesting property?

Update: I stopped the program at $m=407$ since it was taking a long long time to find solution for $m=407$ (if there is any), but all $m=1,\ldots,406$ have a solution.

Update 2: Currently, the program is running the code for approx. half an hour just for $m=490$, and yes, the solutions exist for $m=1,\ldots,489$

Update 3: It seems that it is rare that for some $k \in \mathbb N$ there do not exist some $p$´s such that we do not have $\varphi(pm+k)= \varphi(pm+p+k)$ for all $m$´s, and a research of when that fails to be true seems to be one possible avenue of research.

I am running a program to search for solutions of $\varphi(pm+1)=\varphi(pm+p+1)$.

So far, for $m=1,...,327$ solutions have been found (some relatively large).

(in the body of the question, $p$ is a natural number, not necessarily prime)

I would like to conjecture that for every $m \in \mathbb N$ there exists natural number $p \in \mathbb N$ such that $\varphi(pm+1)=\varphi(pm+p+1)$

• Is this known to be true (or easily deducible)?

• If not known to be true, is there some evidence that totient could have (or not have) this interesting property?

Update: I stopped the program at $m=407$ since it was taking a long long time to find solution for $m=407$ (if there is any), but all $m=1,...,406$ have a solution.

Update 2: Currently, the program is running the code for approx. half an hour just for $m=490$, and yes, the solutions exist for $m=1,...,489$

I am running a program to search for solutions of $$\varphi(pm+1)=\varphi(pm+p+1).$$

So far, for $m=1,\ldots,327$ solutions have been found (some relatively large).

(in the body of the question, $p$ is a natural number, not necessarily prime)

I would like to conjecture that for every $m \in \mathbb N$ there exists natural number $p \in \mathbb N$ such that $\varphi(pm+1)=\varphi(pm+p+1)$

• Is this known to be true (or easily deducible)?

• If not known to be true, is there some evidence that totient could have (or not have) this interesting property?

Update: I stopped the program at $m=407$ since it was taking a long long time to find solution for $m=407$ (if there is any), but all $m=1,\ldots,406$ have a solution.

Update 2: Currently, the program is running the code for approx. half an hour just for $m=490$, and yes, the solutions exist for $m=1,\ldots,489$

I am running a program to search for solutions of $\varphi(pm+1)=\varphi(pm+p+1)$.

So far, for $m=1,...,327$ solutions have been found (some relatively large).

(in the body of the question, $p$ is a natural number, not necessarily prime)

I would like to conjecture that for every $m \in \mathbb N$ there exists natural number $p \in \mathbb N$ such that $\varphi(pm+1)=\varphi(pm+p+1)$

• Is this known to be true (or easily deducible)?

• If not known to be true, is there some evidence that totient could have (or not have) this interesting property?

Update: I stopped the program at $m=407$ since it was taking a long long time to find solution for $m=407$ (if there is any), but all $m=1,...,406$ have a solution.

I am running a program to search for solutions of $\varphi(pm+1)=\varphi(pm+p+1)$.

So far, for $m=1,...,327$ solutions have been found (some relatively large).

(in the body of the question, $p$ is a natural number, not necessarily prime)

I would like to conjecture that for every $m \in \mathbb N$ there exists natural number $p \in \mathbb N$ such that $\varphi(pm+1)=\varphi(pm+p+1)$

• Is this known to be true (or easily deducible)?

• If not known to be true, is there some evidence that totient could have (or not have) this interesting property?

Update: I stopped the program at $m=407$ since it was taking a long long time to find solution for $m=407$ (if there is any), but all $m=1,...,406$ have a solution.

Update 2: Currently, the program is running the code for approx. half an hour just for $m=490$, and yes, the solutions exist for $m=1,...,489$