A225432 Twice the coefficient of sqrt(q) in e^h, where e is the fundamental unit and h is the class number of Q(sqrt(q)), q prime and congruent to 1 mod 4. (The coefficient lies in (1/2)Z, so twice it is an integer.)

1, 1, 2, 1, 2, 10, 1, 5, 250, 106, 1138, 2, 25, 146, 298, 5, 17, 1, 97, 253970, 2, 226, 3034, 9148450, 2050, 10, 157, 126890, 1, 14341370, 5, 110671282, 986, 7586, 530, 130, 173, 5129602, 11068353370, 21685, 694966754, 17883410, 5528222698, 17, 41, 11248618, 60037, 10, 242718010, 24514292738

Offset: 1

Author

Robert R. Bruner, May 07 2013

Keywords

• nonn

Comments

This also arises in the relation satisfied by Euler classes in the connective K-theory of the classifying space of the group of order pq, where p=(q-1)/2. See p. 39 in Bruner and Greenlees, cited below. Take an irreducible representation of the cyclic group of order q which generates the representations as a ring, induce it up to the group of order pq, and let z be its Euler class in ku^{2p}(BG_{pq}). Then z satisfies the relation z^3 -2bq z^2 + qz = 0. This follows from the arithmetic fact that in Q(sort(q)) we have the relation e^h = a + b sqrt(q), as shown on pp. 39-42 of Bruner and Greenlees.

This is closely related to the subsequence of A078357 containing those entries such that the corresponding entry in A077426 is prime. However, a(22) = 226 (corresponding to e^3 = 1710 + 113*sqrt(229)) does not occur in A078357, and more such terms appear after this.

For the n-th Pythagorean prime q=A002144(n), a(n) is also -1/q of the coefficient of term x in the minimal polynomial of A=Product_{a} 2*sin(a*Pi/q) (where the index runs through all quadratic residues in {1,2,...,q-1}) and B=Product_{b} 2*sin(b*Pi/q) (where the index runs through all quadratic nonresidues in {1,2,...,q-1}). It is easy to show that A*B = p. By the class number formula of real quadratic number fields, one obtains B/A = e^(+-2h), so A+B = sqrt(q)*(e^h+e^(-h)) is exactly q*a(n). - Zichang Wang, Dec 15 2022

References

• R. R. Bruner and J. P. C. Greenlees, The Connective K-theory of Finite Groups, Memoirs AMS, Vol. 165, No. 785, 2003.
• T. Mitsuhiro, T. Nakahara and T. Uehara, The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit, Canadian Mathematical Bulletin, 38(1)(1995), 98-103.

Links

• Zichang Wang, Table of n, a(n) for n = 1..2000
• R. R. Bruner and J. P. C. Greenlees, The Connective K-theory of Finite Groups, Semantic Scholar.
• T. Mitsuhiro, T. Nakahara and T. Uehara, The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit.

Crossrefs

Cf. A002144, A077426, A078357.

Programs

Magma

// Magma code to generate all terms for which the prime q is less than or equal to 4N+1 (an initial segment of the sequence). (Note that the brute force computation of the fundamental unit is very inefficient, and will have trouble producing the 39th term.)
N := 40;
pr := [4*n+1 : n in [1..N] | IsPrime(4*n+1)];
for i in [1..#pr] do
   q := pr[i];
   Q := QuadraticField(q);
   h := ClassNumber(Q);
   x := 1;
   while not IsSquare(x*x*q-4) do
      x := x+1;
   end while;
   x := x/2;
   tr,y := IsSquare(x*x*q-1);
   e := y + x*s;
   eh := e^h;
   b := (eh-Trace(eh)/2)/s;
   print i,2*b;
end for;

Mathematica

(* e.g., first 270 terms *)
Lq = Select[4*Range[1000] + 1, PrimeQ[#] &];
Lh = NumberFieldClassNumber[Sqrt[Lq]];
Le = NumberFieldFundamentalUnits[Sqrt[Lq]];
Transpose[RootReduce[(Le^(2 Lh) + 1)/(Sqrt[Lq] Le^Lh)]][[1]]
(* Zichang Wang, Dec 15 2022 *)

Extensions

a(39) onward from Zichang Wang, Dec 15 2022