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The computation below shows that (for $a,b,c \in \mathbb{N}$) the form $$a^2+b^2+c^2+ab+ac+bc$$ covers every odd integer less than $10^5$ except those in $$I= \{ 5, 15, 23, 29, 41, 53, 59, 65, 101, 107, 149, 155, 165, 167, 221, 231, 239, 305, 317, 341, 371, 401, 413, 479, 623, 629, 659, 767, 905, 1001, 1031, 1115, 1397, 1409, 1439, 1751, 1991, 2459, 2543, 4355, 5909, 5969\}.$$

Question: Is it true that the above form covers every odd integer, except those in $I$?

Observation: $2(a^2+b^2+c^2+ab+ac+bc) = (a+b)^2+(a+c)^2+(b+c)^2$.

Application: this answer proves that the form $\| A\|^2$ covers every natural number for $A \in M_3(\mathbb{Z})$.
A positive answer to the above question would prove this result for $A \in M_3(\mathbb{N})$.

Computation

sage: oddmixed(317)
{5, 15, 23, 29, 41, 53, 59, 65, 101, 107, 149, 155, 165, 167, 221, 231, 239, 305, 317, 341, 371, 401, 413, 479, 623, 629, 659, 767, 905, 1001, 1031, 1115, 1397, 1409, 1439, 1751, 1991, 2459, 2543, 4355, 5909, 5969}

Code

# %attach SAGE/3by3.spyx

from sage.all import *

cpdef oddmixed(int r):
    cdef int a,b,c,n,i
    cdef list L
    L=[]
    for a in range(r):
        for b in range(r):
            for c in range(r):
                n=a**2+b**2+c**2+a*b+a*c+b*c
                if not n in L:
                    L.append(n)
    return set([2*i+1 for i in range(r*r/2)])-set(L)
Sebastien Palcoux
  • 5,797
  • 2
    An equivalent formulation: If $n\equiv 2\pmod 4$ and $n>45000$, then $n=a^2+b^2+c^2$ where $a,b,c$ are integer sides of a triangle (i..e, $0<a\le b\le c<a+b$) – Hagen von Eitzen Oct 28 '18 at 19:01
  • @HagenvonEitzen: Yes! If $(x,y,z)$ are integer sides of a triangle and $x+y+z \equiv 0 \pmod 2$, then for $2a=x+y-z$, $2b=x-y+z$, $2c=-x+y+z$, we have $a,b,c \in \mathbb{N}$, $x=a+b$, $y=a+c$, $z=b+c$, so that $x^2+y^2+z^2 = (a+b)^2+(a+c)^2+(b+c)^2 = 2(a^2+b^2+c^2+ab+ac+bc)$. Thanks to the last checking, we can assume $n>80000$. – Sebastien Palcoux Oct 28 '18 at 21:59

0 Answers0