For every prime power q, every integer e≥ 2, every integer r≥ 1, and every a∈F_q^e, the binomial f(X)=X^r bigl(X^q-1+a bigr)∈F_q^e[X] is a permutation polynomial of F_q^e if and only if, writing ell_j(q)=fracq^j-1q-1, one has gcd(r,q-1)=1, (-a)^ell_e(q)≠ 1, and there exists an integer h with 1≤ h<e and gcd(h,e)=1 such that r ell_h(q)equiv 1 pmodell_e(q).

Masuda–Rubio–Santiago conjecture

For every prime power $q$ , every integer $e\ge 2$ , every integer $r\ge 1$ , and every $a\in\mathbb{F}_{q^e}$ , the binomial $f(X)=X^r\bigl(X^{q-1}+a\bigr)\in\mathbb{F}_{q^e}[X]$ is a permutation polynomial of $\mathbb{F}_{q^e}$ if and only if, writing $\ell_j(q)=\frac{q^j-1}{q-1}$ , one has $\gcd(r,q-1)=1$ , $(-a)^{\ell_e(q)}\ne 1$ , and there exists an integer $h$ with $1\le h<e$ and $\gcd(h,e)=1$ such that $r\ell_h(q)\equiv 1\pmod{\ell_e(q)}$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Structural composition formulation
Every permutation binomial of the form $X^r\bigl(X^{q-1}+a\bigr)$ over $\mathbb{F}_{q^e}$ arises, for some $h$ with $1\le h<e$ and $\gcd(h,e)=1$ , from the composition $\bigl(X^{q^h}+aX\bigr)\circ X^r$ .
source: A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields

References

Primary source

arXiv

Additional references

A complete classification of permutation binomials of the form X^r(X^{q-1}+a) over finite fields — arXiv — Xiang Fan

finite-fields number-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new preprint claims to settle the conjecture by classifying the relevant finite-field permutation polynomials, but the result is unrefereed and unverified.

The Masuda–Rubio–Santiago conjecture concerns a complete structural classification of a significant family of finite-field permutation polynomials. No proposer or original date is identified in the retrieved material.

September 17, 2026 claimed proof

Xiang Fan’s preprint A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields claims necessary and sufficient conditions, counts the resulting permutation functions, and extends the classification to a broader coprime-index family. If correct, this proves the conjecture, but the preprint is newly posted and unrefereed.

Current status (as of September 2026): The conjecture has a claimed complete solution in Xiang Fan’s September 17 preprint, but its correctness remains unverified.

Sources

No solutions have been posted yet.

Structural composition formulation

Primary source

Additional references

Progress summary