The upper-bound conjecture for the Ova-Ova-Prime-Ova construction

About 5 years old · traced to 2021

Let the preceding Ova-Ova-Prime-Ova construction be given, and let $N$ denote the number of primes it produces. The Ova-Ova-Prime-Ova counting conjecture. Although the preceding theorem guarantees at least one resulting prime, one has $N\leq 10$ . The source supplies examples and asserts this as a bound, but gives no independent resolution evidence.

References

Primary source

Yeisson Alexis Acevedo Agudelo, “Prime numbers. An alternative study using ova-angular rotations”, arXiv:2104.04522 (2021).

elementary-number-theory prime-numbers

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A posted calculation claims the ten-output limit is false by producing $38$ outputs from one pair of input primes, but nobody has independently verified it.

The conjecture asserts that the Ova-Ova-Prime-Ova construction produces at most $10$ primes. Yeisson Alexis Acevedo Agudelo introduced the underlying construction in a 2021 preprint, which records an existence result but does not independently settle this upper bound.

Known results

Acevedo Agudelo (2021) gives the construction and guarantees at least one resulting prime, while supplying examples supporting the proposed bound $N\le 10$ .

Posted attempt

A reader supplies two input primes, $677$ and $401$ , and lists $38$ residues satisfying the construction’s primality conditions, claiming $N=38$ and thereby refuting both $N\le 10$ and the earlier bound $N\le 13$ . The calculation has not been independently verified.

Current status (as of August 2026): The published construction and at-least-one-output result are recorded, while the ten-output conjecture has a specific but unverified claimed counterexample with $N=38$ .

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

Take the primes

\rho_1=677=317+360\cdot1, \qquad \rho_2=401=41+360\cdot1.

Their residues are $k_{\rho_1}=317$ and $k_{\rho_2}=41$ , and both rotation frequencies are $1$ . Thus a residue $r$ contributes an output of the prescribed construction precisely when

\gcd(r,360)=1,\qquad k_{\rho_1}+k_{\rho_2}+2-r=360-r\ \text{is prime}, \qquad r+360(1+1)=720+r\ \text{is prime}.

The following $38$ distinct residues satisfy all three conditions:

\begin{aligned} r\in\{& 7,13,23,49,53,67,77,89,91,103,109,119,133,137,161,163,167,187,209,\\ &221,233,247,251,257,263,271,277,289,293,299,301,313,319,329,331,341,343,349 \}. \end{aligned}

For example, the first eleven give

\begin{array}{c|c|c} r&360-r&720+r\\\hline 7&353&727\\ 13&347&733\\ 23&337&743\\ 49&311&769\\ 53&307&773\\ 67&293&787\\ 77&283&797\\ 89&271&809\\ 91&269&811\\ 103&257&823\\ 109&251&829 \end{array}

and every entry in the last two columns is prime.

Consequently this single pair of input primes produces $N=38$ prime outputs, contradicting the stated bound $N\le10$ . The earlier published version of the construction proposed the weaker bound $N\le13$ ; the same example disproves that original bound as well.