We study equations in
which the unknowns are the exponents. (Work in this field originated with
C. Størmer and D. H. Lehmer. More recently, Leo J. Alex has extended
their results; his work relates to classification of nonabelian simple groups.)
For the equation k + 7α= 3c+ 5d, k = 3b,5b,13b, or 17b, and for many
similar 4-term equations, we find all integral solutions.
We find all integral solutions of 3α+ 7b= 3c+ 5d+ 2.
We prove that there are infinitely many odd m such that mα+7b= 3c+5d
has only the solutions (a,b,c,d) = (0,0,0,0),(0,1,1,1).
