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David Wells, in his entertaining but non-scholarly Curious and Interesting Numbers (1986, 2 ed. 1997) reports that the positive integer $1 \, 375 \, 298 \, 099$ can be expressed as the sum of $3$ fifth powers in $2$ different ways:

$$1 \, 375 \, 298 \, 099 = 24^5 + 28^5 + 67^5 = 3^5 + 54^5 + 62^5$$

He gives the source as "R. Alter" but I can find no reference to this person.

I have seen a suggestion that his first name may be Roland, but I can't find any justification for this.

(MathWorld has a page on this: https://mathworld.wolfram.com/DiophantineEquation5thPowers.html in which it is stated that this is the smallest number with this property.)

Can anyone help me out with any information about R. Alter?

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  • $\begingroup$ @njuffa Thanks, I'll research those. $\endgroup$ Aug 26, 2022 at 10:22
  • 3
    $\begingroup$ One might think that the OP would learn how to do these investigations themselves after a few of them… $\endgroup$
    – Jon Custer
    Aug 26, 2022 at 13:30

1 Answer 1

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The particular dual sum-of-powers representations of $1375298099$ from the question appear in Table 4 of

Ronald Alter, "Computations and generalizations on a remark of Ramanujan." In: Marvin I. Knopp (ed.), Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980 (Lecture Notes in Mathematics, 899; Springer 1981).

This table is on page 190. The conference was organized on the occasion of the retirement of Alter's Ph.D. thesis advisor Emil Grosswald from Temple University.

The earliest mention of Ronald Alter in the literature I could find is in a list of new members of the American Mathematical Society in the following publication:

John W. Green and Morton L. Curtis, "The Annual meeting in Miami", Bulletin of the American Mathematical Society, Vol. 70, 1964, pp. 343-356 (scan). On page 351:

University of Pennsylvania: Ronald Alter, Robert H. Cantor, [...]

Per his record at the Mathematics Genealogy Project, Ronald Alter received his Ph.D. in 1965 from the University of Pennsylvania. The title of his Ph.D. thesis is: On a Necessary Condition for the Validity of the Riemann Hypothesis for Functions that Generalize the Riemann Zeta Function. A publication based on the Ph.D. thesis appeared in 1968; Alter's affiliation is stated as University of California, Los Angeles.

Ronald Alter, "On a necessary condition for the validity of the Riemann hypothesis for functions that generalize the Riemann zeta function." Transactions of the American Mathematical Society, Vol. 130, No. 1, 1968, pp. 55-74 (scan).

After receiving his Ph.D., Alter briefly joined the faculty of UCLA. General Catalog University of California, Los Angeles, Regents of the University of California 1966, p. 319:

Ronald Alter, Ph.D., Acting Assistant Professor of Mathematics.

Two papers published in 1968 state Alter's affiliation as System Development Corporation, Santa Monica, California. SDC was one of the first computer software companies, founded in 1955. The first of these two papers states that Alter is a postdoctoral fellow.

Ronald Alter, "On the Nonexistence of Close-Packed Double Hamming-Error-Correcting Codes on q = 7 Symbols", Journal of Computer and System Sciences, Vol. 2, No. 2, Aug. 1968, pp. 169-176

Ronald Alter, "On the Non-Existence of Perfect Double Hamming-Error-Correcting Codes on q = 8 and q = 9 Symbols", Information and Control, Vol. 13, No. 6, Dec. 1968, pp. 619-627.

Ronald Alter joined the faculty of the University of Kentucky in 1970. In Notices of the American Mathematical Society, Volume 17, Number 5, August 1970, p. 735:

Professor RONALD ALTER of System Development Corporation has been appointed to an associate professorship at the University of Kentucky.

The American Mathematical Monthly, Vol. 77, 1970, p. 1035:

Assistant Professor Ronald Alter, University of Kentucky, has been promoted to Associate Professor

I cannot find any publication by Alter after 1980, or in fact any other information on him after that year. All publications from 1970 through 1980 state his affiliation as University of Kentucky, and later ones more specifically mention the Department of Computer Science.

In online information I found people called Ronald Alter, e.g. multiple ones born between 1934 and 1939 in the 1940 United States Census, or one in a 1958 highschool yearbook, but it is impossible to determine at this time whether any of them are identical to the mathematician Ronald Alter. I have not been able to find a relevant obituary.


Other publications by Ronald Alter:

Ronald Alter, "Perfect Double Hamming-Error-Correcting Codes on Q-Symbols", System Development Corporation, Mar. 1969 (DTIC AD0685436)
Ronald Alter, "Perfect Double Hamming-Error-Correcting Codes on Q-Symbols", Proceedings of the Third Annual Princeton Conference on Information Sciences and Systems, March 27-28, 1969, pp. 547-550.
Ronald Alter and B. P. Lientz, "A Generalization of a Combinatorial Problem of Smirnov", System Development Corporation, SP-3254, Nov. 1968 (also: DTIC AD0680108)
Ronald Alter and B. P. Lientz. "Applications of a generalized combinatorial problem of Smirnov." Naval Research Logistics Quarterly, Vol. 16, No. 4, Dec. 1969, pp. 543-547
Ronald Alter and Mark Villarino, "A remark on primes in arithmetic progressions", Bulletin of the Australian Mathematical Society, Vol. 3, No. 2, Oct. 1970, pp. 185-192 (scan)
Ronald Alter and Bennet Lientz, "A note on a problem of Smirnov - A graph theoretic interpretation", Naval Research Logistics Quarterly, Vol. 17, No. 3, Sep. 1970, pp. 407-408 (scan)
Ronald Alter and Thaddeus B. Curtz, "On binary nonassociative products and their relation to a classical problem of Euler", University of Kentucky, Computer Science Dept., 1970 (?)
Ronald Alter, "On a Diophantine equation related to perfect codes", Mathematics of Computation, Vol. 25, No. 115, July 1971, pp. 621-624 (scan)
Ronald Alter and Thaddeus B. Curtz, "An Experimental Computer Science Problem Seminar", ACM SIGCSE Bulletin, Vol. 4, No.1, Mar. 1972, pp. 144-153 (scan)
Ronald Alter, "Five Full Houses!", problem 820, Mathematics Magazine, Vol. 45, No. 5, Nov. 1972, p. 289 (scan)
Ronald Alter and Bennet P. Lientz. "Remarks on a combinatorial problem in orderings." Sankhyā: The Indian Journal of Statistics, Series A, Vol. 34, No. 4, Dec. 1972, pp. 457-460.
Ronald Alter, "Can $\phi(n)$ properly divide $n-1$", The American Mathematical Monthly, Vol. 80, No. 2, Feb. 1973, pp. 192-193
Ronald Alter and Thaddeus B. Curtz, "On binary non-associative products and their relation to a classical problem of Euler", Commentationes Mathematicae, Vol 17, No. 1, 1973, pp. 1-8 (scan)
Ronald Alter and K. K. Kubota, "Multiplicities of second order linear recurrences", Transactions of the American Mathematical Society, Vol. 178, Apr. 1973, pp. 271-284 (scan)
Ronald Alter and K. K. Kubota, "The diophantine equation $x^{2}+D=p^{n}$," Pacific Journal of Mathematics, Vol. 46, No. 1, November 1973, pp. 11-16 (scan)
Ronald Alter and K. K. Kubota, "Prime and prime power divisibility of Catalan numbers", Journal of Combinatorial Theory, Series A, Vol. 15, No. 3, Nov. 1973, pp. 243-256 (scan)
Ronald Alter and Thaddeus B. Curtz, "A note on congruent numbers", Mathematics of Computation, Vol. 28, No. 125, Jan. 1974, pp. 303-305 (scan)
Ronald Alter and K. K. Kubota, "The Diophantine equation $x^{2}+11=3^{n}$ and a related sequence", Journal of Number Theory, Vol. 7, No. 1, Feb. 1975, pp. 5-10 (scan)
Ronald Alter, "How many Latin squares are there?", The American Mathematical Monthly, Vol. 82, No. 6, Jun. 1975, pp. 632-634
Ronald Alter, "A Characterization of Self-Complementary Graphs of Order 8", Portugaliae Mathematica, Vol. 34, Fasc. 3, 1975, pp. 157-161 (scan)
"Correction to:“A note on congruent numbers”(Math. Comp. 28 (1974), 303–305)", Mathematics of Computation, Vol. 30, No. 133, 1976, p. 198 (scan)
Ronald Alter and Chung C. Wang, "Uniquely Intersectable Graphs", Discrete Mathematics, Vol. 18, No. 3, 1977, pp. 217-226 (scan)
Ronald Alter and Jeffrey A. Barnett, "Remarks on the postage stamp problem with applications to computers", University of Kentucky, Department of Computer Science, 1977 (scan)
Ronald Alter and Johnson M. Hart, "Enumerating syntactical graphs and lattices of derivations", International Journal of Computer & Information Sciences, Vol. 8, No. 4, Aug. 1979, pp. 261-277
Ronald Alter, "The congruent number problem", The American Mathematical Monthly, Vol. 87, No. 1, Jan. 1980, pp. 43-45
Ronald Alter and Jeffrey A. Barnett, "A Postage Stamp Problem", The American Mathematical Monthly, Vol. 87, No. 3, Mar. 1980, pp. 206-210

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