Abstract:
Letkanddbe a given positive integers and suppose thataris a given sequence. In this current study, we investiagate adiophantine identity relating the sums of squares and quartic from specific sequences to a variabled.In particular, thediophantine identity∑2kr=1a4r+kd4=2∑2k−1r=1(arar+1+d2)2is developed and introduced. The objective of this researchis to determine the conditions under which integer solutions for(ar,d)exist within this diophantine equation. Themethodology involves, decomposing polynomials, factorizing polynomials, and exploring the solution set of the givenequation.
