ACTA issues

A new conjecture on integer powers

Susil Kumar Jena

Acta Sci. Math. (Szeged) 81:3-4(2015), 425-430

Abstract. We make a conjecture about integer powers which states that for any integer $n\geq2$, the $n^{th}$ power of any arbitrary integer, including zero, can be expressed `primitively' and `non-trivially', in infinitely many different ways as the sum or difference of $(n + 1)$ number of other non-zero, but not necessarily distinct integral $n^{th}$ powers. The conjecture is established for squares, cubes (partly) and biquadrates, and is open for the remaining cases. Finally, a few more questions are raised for further investigation.

DOI: 10.14232/actasm-013-319-2

AMS Subject Classification (1991): 11D41, 11P05

Keyword(s): diophantine equation, conjecture on integer powers (coip), Waring-type problems

Received October 10, 2013, and in revised form November 3, 2014. (Registered under 69/2013.)