Expressing GCD as a linear combination where coefficients are non zero

Question

Let $n_1,n_2,\cdots n_k$ be $k$ natural numbers and $a_1,a_2,\cdots,a_k$ be integers such that $$gcd(n_1,n_2,\cdots,n_k)=a_1n_1+a_2n_2+\cdots+a_kn_k = \sum_{i=1}^{k}a_in_i$$ .GCD can always be expressed as a linear combination of $n_{i's}$(proof in comment), but my objective is to find natural numbers $n_i$ such that all of the $a_{i's}$ are non zero.

For $k=3$, $\text{gcd}(6,15,77)=6(102)+15(-51)+77(2) = 1$ will be a valid example, but $\text{gcd}(2,3,5)=2(-1)+3(1)+5(0)=1$ is not a valid example as $a_3=0$. Similarly, for $k=4$ , $\text{gcd}(6,15,35,77)=6*(46)+15(-23)+35(2)+77(0) = 1$ will not be a valid example as $a_4=0$, but $\text{gcd}(210,510,2805,10210)=210(-1518876)$ $+510(632865)$ $+2805(-1361)$ $+10210(2) = 5$ is a valid example. I was also able to find an example for $k=5$, $\text{gcd}(210,510,2805,10210,102102)$ $=210(93047862636)$$+510(-38769942765)$ $+2805(83376221)$$+10210(-122522)$$+102102(3) = 1$. I couldn't find an example for $k=6$.

I believe that for every natural number $k$, natural numbers $n_i$ exist which satisfy this property. But I am not able to find a proof or construct such numbers. For a particular $k$, let $(n_1,n_2,\cdots,n_k)$ and $(m_1,m_2,\cdots ,m_k)$ be two pairs satisfying this property. Define$(n_1,n_2,\cdots ,n_k) < (m_1,m_2,\cdots,m_k)$ if $\sum_{i=1}^{k} n_i < \sum_{i=1}^{k} m_i$ and they are equal if their sums are equal. Then for $k=2$ , $(2,3)$ is the smallest solution. For $k=3$ i believe it will be $(6,15,35)$, $\text{gcd}(6,15,35)=6(46)$$+15(-23)+35$$(2)=1$, but i don't have a proof for this. What will be the samllest solutions for other values of $k$?

Answer 1

Suppose Bezout is $\,d\, =\, 0\cdot n_1 +\, \cdots\, + 0\cdot n_j\, +\, a_{j+1}\, n_{j+1} + \cdots + a_k\, n_k,\, $ all $\,a_i \neq 0,\ j\ge 1$.

Choose $a_1\!\neq 0\,$ so $\,1 + a_1 n_1 + n_2 + \cdots + n_j =: c \ne 0.\ $ Multiply above Bezout equation by this

which yields that $\, d + d a_1 n_1\! + dn_2 + \cdots + dn_j =\, c a_{j+1}\, n_{j+1} + \cdots + c a_k\, n_k\,$ with all coef's $\neq 0$.

Answer 2

The standard Bézout's identity for $2$ integers, plus the Bézout's identity generalization to $3$ or more integers, confirms there exists integers $a_i, \; 1 \le i \le k$, such that

$$d=\gcd(n_1,n_2,\cdots,n_k)=\sum_{i=1}^{k}a_in_i \tag{1}\label{eq1A}$$

for any $k \ge 2$. First, consider the case where $k = 2$ to give

$$d = a_1 n_1 + a_2 n_2 \tag{2}\label{eq2A}$$

Both $a_1$ and $a_2$ cannot be $0$. Suppose one of them, WLOG $a_1$, is $0$. Then you can select new coefficients $a_1^{'} = a_1 + b\left(\frac{n_2}{d}\right)$ and $a_2^{'} = a_2 - b\left(\frac{n_1}{d}\right)$ for any integer $b$. For any $b \neq 0$ you have $a_1^{'} \neq 0$ and, for all but possibly $1$ value of $b$, you have $a_2^{'} \neq 0$, showing you can choose a $b$ to get $a_1^{'} \neq 0$ and $a_2^{'} \neq 0$.

I'm going to show by strong induction and construction you can always find for any $k \ge 2$ a set of all non-zero $a_i$ coefficients. The base case for $k = 2$ has been shown in the paragraph above. Assume it's true for all $k \le c$ for some integer $c \ge 2$. Consider $k = c + 1$. If all of the $a_i$ are non-zero, then you're done. Otherwise, since not all of the $a_i$ may be zero, you have some $e$, where $1 \le e \le c$, values of $a_i$ which are $0$.

Let $f_i$ for $1 \le i \le e$ be the indices of $a$ and $n$ where the coefficients are $0$. There are $2$ basic cases to consider.

Case 1:

If $e \ge 2$, then by the induction assumption there exists coefficients, call them $g_i ,\; 1 \le i \le e$, which are all are non-zero. Also, have $h$ be the $\gcd$ of the corresponding $n_i$ values. Note that $d \mid h$, say $h = md$ for some integer $m \ge 1$. Also, consider any integer $q \ge 2$. You then have

$$\sum_{i=1}^{e} (q)(g_i)(n_{f_i}) = q(h) = (qm)d \tag{3}\label{eq3A}$$

Now, consider \eqref{eq3A} minus $qm - 1$ times \eqref{eq1A}. The resulting value is $d$. Also, due to no overlap between the coefficients of the $2$ equations, all of the original zero coefficients are now the corresponding non-zero values of $(q)(g_i)$, plus the original non-zero coefficients are now the non-zero $-(qm-1)(a_i)$ values. Thus, all of the coefficients are now non-zero.

Case 2:

Consider $e = 1$. As before, have $f_1$ be the zero coefficient index. Choose any other coefficient and have $f_2$ be the index of that value. Use this to construct \eqref{eq3A} and proceed as before there. If the result has the other chosen coefficient to be non-zero, then you're done. Otherwise, increase $q$ by $1$, with this then causing the resulting coefficient to decrease by $m(a_i)$ and, thus, be non-zero. As before, the end result is a set of all non-zero coefficients.

Thus, coefficients can be chosen in all cases so it's true for $k = c + 1$ as well. This completes the induction procedure to show you can always choose non-zero coefficients.

Update: I later realized I could have started at $k = 1$, using a definition of $\gcd(n_1) = n_1$, to give that $a_1 = 1$ since $(1)n_1 = n_1$. This would have made the above induction proof shorter & simpler. Alternatively, I could have started as above but then, for Case 2, used just the one zero coefficient value with $h = n_1$ for use in the Case 1 procedure.