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This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $a{x}^k+b{y}^k+c{z}^k\equiv d\left(modp\right)$ has a solution with $xyz\not\equiv 0\left(modp\right)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2·3^{R-1}·k+1$ variables, has always non-trivial $p$-adic solutions, provided $p\nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p\>k^4$.

In this paper, we find all solutions of the Diophantine equation $x^2+2^{\alpha }{5}^{\beta }{17}^{\gamma }=y^n$ in positive integers $x,y\ge 1$, $\alpha ,\beta ,\gamma ,n\ge 3$ with $gcd(x,y)=1$.