What the study found
The study shows that for odd prime numbers p < q, the second gap of the binary cyclotomic polynomial Φpq is the maximum of r − 1 and p − r − 1, where r is the remainder of q divided by p.
Why the authors say this matters
The authors also state that, when q is congruent to ±1 modulo p, they give the number of gaps for each possible length. They present this as part of a new way to describe the coefficients of binary cyclotomic polynomials.
What the researchers tested
The researchers studied binary cyclotomic polynomials of order pq, where p and q are odd primes. They developed a new approach in which the coefficients are described as concatenations of words arising from iterations of a circular map.
What worked and what didn't
Their approach led to a proof of the formula for the second gap of Φpq. It also allowed them, in the case q ≡ ±1 mod p, to determine the number of gaps for each possible length.
What to keep in mind
The abstract does not describe limitations or exceptions beyond the stated conditions on p and q. The results are presented only for odd prime numbers p < q and for the specific case q ≡ ±1 modulo p when counting gaps by length.
Key points
- For odd primes p < q, the second gap of Φpq equals max(r − 1, p − r − 1).
- Here r is the remainder when q is divided by p.
- When q ≡ ±1 (mod p), the authors give the number of gaps for each possible length.
- The paper introduces an approach based on concatenations of words from iterations of a circular map.
- The abstract does not describe limitations beyond the stated prime-number conditions.
Disclosure
- Research title:
- Second gap of binary cyclotomic polynomials determined
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