[math-fun] Numbers of the form K^2 + KL + L^2
Numbers of the form K^2 + KL +L^2, K and L integers, arise often when making calculations connected with the triangular lattice in the plane. Let tau = exp(2pi*i/6). For K, L > 0, K^2 + KL +L^2 is, among other things, the number of points in the quotient ring Z{tau] / <K + L*tau> namely, the Eisenstein integers factored out by its ideal <K + L*tau>. I would like to know more about these numbers. For example: * Given any integer N, is there a simple test for whether N is of the form K^2 + KL + L^2 for some integers K, L ??? * What can be said about the prime factorization of K^2 + KL + L^2 ??? * Is there a relationship of such numbers to Eisenstein primes? —Dan
At OEIS, http://oeis.org/A003136 gives a bunch of information about these numbers. On Wed, Jul 20, 2016 at 4:11 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Numbers of the form K^2 + KL +L^2, K and L integers, arise often when making calculations connected with the triangular lattice in the plane.
Let tau = exp(2pi*i/6).
For K, L > 0, K^2 + KL +L^2 is, among other things, the number of points in the quotient ring
Z{tau] / <K + L*tau>
namely, the Eisenstein integers factored out by its ideal <K + L*tau>.
I would like to know more about these numbers.
For example:
* Given any integer N, is there a simple test for whether N is of the form K^2 + KL + L^2 for some integers K, L ???
* What can be said about the prime factorization of K^2 + KL + L^2 ???
* Is there a relationship of such numbers to Eisenstein primes?
—Dan
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The discussion for A003136 mentions that the set of such numbers is closed under multiplication. Some minor unmentioned facts: The set is also closed under restricted division: If M and N are members, and M divides N, then N/M is a member. If N=2(mod3), N is not in the sequence. The density of members (relative to the integers>0) gradually falls to 0. IIRC, the density goes as O(1/sqrt(log N)). This implies that, if N is a member, the average expected number of representations of N is O(sqrt(log N)). Representations usually come in sets of 6: (K,L), (K+L,-K), (K+L,-L) and their negatives. Rich ---- Quoting Allan Wechsler <acwacw@gmail.com>:
At OEIS, http://oeis.org/A003136 gives a bunch of information about these numbers.
On Wed, Jul 20, 2016 at 4:11 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Numbers of the form K^2 + KL +L^2, K and L integers, arise often when making calculations connected with the triangular lattice in the plane.
Let tau = exp(2pi*i/6).
For K, L > 0, K^2 + KL +L^2 is, among other things, the number of points in the quotient ring
Z{tau] / <K + L*tau>
namely, the Eisenstein integers factored out by its ideal <K + L*tau>.
I would like to know more about these numbers.
For example:
* Given any integer N, is there a simple test for whether N is of the form K^2 + KL + L^2 for some integers K, L ???
* What can be said about the prime factorization of K^2 + KL + L^2 ???
* Is there a relationship of such numbers to Eisenstein primes?
?Dan
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Since Q(zeta), where zeta is a primitive 3rd root of unity has class number 1, the situation as to whether an integer is of the form x^2 + xy + y^2 is similar to the situation with x^2 + y^2: n is of the that form if and only if every prime p dividing n which is = 5 mod 6 divides it to an even power. The density of of 1/sqrt(x) that Rich mentioned is an old result due to Landau. Victor On Wed, Jul 20, 2016 at 4:11 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Numbers of the form K^2 + KL +L^2, K and L integers, arise often when making calculations connected with the triangular lattice in the plane.
Let tau = exp(2pi*i/6).
For K, L > 0, K^2 + KL +L^2 is, among other things, the number of points in the quotient ring
Z{tau] / <K + L*tau>
namely, the Eisenstein integers factored out by its ideal <K + L*tau>.
I would like to know more about these numbers.
For example:
* Given any integer N, is there a simple test for whether N is of the form K^2 + KL + L^2 for some integers K, L ???
* What can be said about the prime factorization of K^2 + KL + L^2 ???
* Is there a relationship of such numbers to Eisenstein primes?
—Dan
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participants (4)
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Allan Wechsler -
Dan Asimov -
rcs＠xmission.com -
Victor Miller