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We prove that for each n ∈ ℕ₊ the Diophantine equation A² ± nB⁴ + C⁴ = D⁸ has infinitely many primitive integer solutions, i.e. solutions satisfying gcd(A,B,C,D) = 1.

"Ramanujan's 6-10-8 identity" inspired Hirschhorn to formulate his "3-7-5 identity". Now, we give a new "6-14-10 identity" which we suppose Ramanujan would have discovered but missed to mention in his notebooks.

The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.

In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for $(A,B,C)$ of the Diophantine equation $-(2^p·A^6)+B^3=C^2$ for any positive integral values of $p$ when $p\equiv 1$ (mod 6) or $p\equiv 2$ (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.

Each of the Diophantine equations $A^4\pm n{B}^3=C^2$ has an infinite number of integral solutions $(A,B,C)$ for any positive integer $n$. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$, $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A,B,C)$ of the Diophantine equation $a{A}^3+c{B}^3=C^2$ for any co-prime integer pair $(a,c)$.

In p. 219 of R.K. Guy’s , 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^n+y^n=n!z^n$ has no integer solutions with $n\in {\mathbb{N}}_{+}$ and $n>2$. But, contrary to this expectation, we show that for $n=3$, this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition $gcd(x,y,z)=1$.

The two related Diophantine equations: $X_1^4+4X_2^4=X_3^8+4X_4^8$ and $Y_1^4=Y_2^4+Y_3^4+4Y_4^4$, have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.