On Trinomial Equations in a Finite Field.

where g is a multiplicative generator of the nonzero elements of F [pfn; i is a fixed integer in the set 0, 1, . . ., c 1; j is a fixed integer in the set 0, 1, .m. ., 1; pf _ 1 = cm; s isin the range 0, 1, . . ., m -1; t is in the range 0, 1, ..., c 1. Let the number of sets of solutions s, t of the above type be (i, j),m. Then (noting relation (2) of the present paper) the maximum value for (i, j),m is m if c > m. If there exists a j for a fixed i such that (i, j)cm = m, we shall call this Case A of (1), and if i is fixed and (i, j)cr = 1 for each j in the set 0, 1, . . ., m 1, we shall call this Case B of (1). In a recent paper1 Mrs. E. H. Pearson and the writer considered relation (1) for n = 1, equivalent to using congruences modulo p in (1), where we used, first, the less convenient terms and, second, extreme cases in lieu of Cases A and B (Paper T, pp. 1284-1285). In Paper T problems of this character were, as far as I know, considered for the first time in the literature. However, the writer had been working on such questions for some years previously but did not publish his results. As it may be possible some time not too far in the future to check various criteria that we have obtained, on the high-speed digital computing machines, we shall attempt to publish from time to time material along this line and shall not wait until the theory is extensively developed before publication of this material. Again in Paper T (p. 1281) an application to diophantine analysis was given; that is, a number of classes of diophantine equations were shown to be impossible in rational integers. Also, in a second paper2 a connection of Case A with the Fermat problem was revealed. In the present paper we set up a number of formulas which we shall in later papers use to obtain more definite information about the (i, j)'s, with particular attention to Cases A and B. Hence we try to find, if possible, relations and criteria involving only one of the (i, j). We approach that in the present article in the criteria (12) involving in each relation only pairs of (i, j)'s. 1. Now consider the relation (1), with the number m, an integer >2. We note that rnM 1 m ifi_0, (2) _(i0) = I 2 -0 ~ 1ifi=0.