Squarefree values of trinomial discriminants

@article{Boyd2014SquarefreeVO,
  title={Squarefree values of trinomial discriminants},
  author={David W. Boyd and Greg Martin and Mark Thom},
  journal={Lms Journal of Computation and Mathematics},
  year={2014},
  volume={18},
  pages={148-169}
}
The discriminant of a trinomial of the form x n x m 1 has the form n n (n m) n m m m if n and m are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, whenn is congruent to 2 (mod 6) we have that ((n 2 n+1)=3) 2 always divides n n (n 1) n 1 . In addition, we discover many other square factors of these discriminants that do not t into these… 

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References

SHOWING 1-10 OF 24 REFERENCES

A construction of polynomials with squarefree discriminants

For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients

An estimate for Heilbronn's exponential sum

for any integer a coprime to p. It is important to note here that if n ≡ n′ (mod p), then n ≡ n′p (mod p). Thus the summand in S(a) has period p with respect to n, so that S(a) is a ‘complete sum’ to

Factorization of polynomials over finite fields.

Dickson [1, Ch. V, Th. 38] has given an interesting necessary condition for a polynomial over a finite field of odd characteristic to be irreducible. In Theorem 1 below, I will give a generalization

On Cauchy–Liouville–Mirimanoff Polynomials

  • P. Tzermias
  • Mathematics
    Canadian Mathematical Bulletin
  • 2007
Abstract Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy–Liouville–Mirimanoff polynomials to show that the intersection of

The new book of prime number records

1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.

Old and new conjectured diophantine inequalities

The original meaning of diophantine problems is to find all solutions of equations in integers or rational numbers, and to give a bound for these solutions. One may expand the domain of coefficients

Wieferich's criterion and the abc-conjecture

13 lectures on Fermat's last theorem

Lecture I The Early History of Fermat's Last Theorem.- 1 The Problem.- 2 Early Attempts.- 3 Kummer's Monumental Theorem.- 4 Regular Primes.- 5 Kummer's Work on Irregular Prime Exponents.- 6 Other

On the Oesterlé-Masser conjecture

Letx, y andz be positive integers such thatx=y+z and ged (x,y,z)=1. We give upper and lower bounds forx in terms of the greatest squarefree divisor ofx y z.