Abstract
In this work, we solve the Diophantine equations \( p^x+q^y=z^3 \) in the set of nonnegative integers. We consider twin primes \( p \text { and } q\) having the following forms:
-
(a)
twin primes of the form \( 8N+1 \) and \( 8N+3 \) for some \( N\in \mathbb {N}_0\),
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(b)
twin primes of the form \( 8N+3 \) and \( 8N+5 \) for some \( N\in \mathbb {N}_0\),
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(c)
twin primes of the form \( 8N+5 \) and \( 8N+7 \) for some \( N\in \mathbb {N}_0\), and
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(d)
twin primes of the form \( 8N+7 \) and \( 8N+9 \) for some \( N\in \mathbb {N}_0\).
Theorems are established. Conjectures are provided for further studies.
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Acknowledgements
The authors would also like to convey their gratitude to Dr. Jerome Dimabayao and Dr. Julius Fergy Rabago for their assistance in polishing some proofs for this research study. The authors would also like to thank the University of the Philippines for its support in disseminating the results and in publishing the manuscript.
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Aquino, R.L., Bacani, J.B. (2022). On the Exponential Diophantine Equation \( p^x+q^y=z^3: \) Theorems and Conjectures. In: Giri, D., Raymond Choo, KK., Ponnusamy, S., Meng, W., Akleylek, S., Prasad Maity, S. (eds) Proceedings of the Seventh International Conference on Mathematics and Computing . Advances in Intelligent Systems and Computing, vol 1412. Springer, Singapore. https://doi.org/10.1007/978-981-16-6890-6_52
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