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Copyright Notice

The following manuscript
          EWD 650 A theorem about odd powers of odd integers
is held in copyright by Springer-Verlag New York.
The manuscript was published as pages 349350 of
Edsger W. Dijkstra, Selected Writings on Computing: A Personal Perspective,
Springer-Verlag, 1982. ISBN 0387906525.
Reproduced with permission from Springer-Verlag New York.
Any further reproduction is strictly prohibited.

A theorem about odd powers of odd integers.

Theorem. For any odd p ≥ 1 , integer K ≥ 1 , and odd r such that that 1 ≤ r < 2K, a value x exists such that

R:1 ≤ x < 2K and 2K|(xp-r) and odd(x) .
Note. For “a|b” read: “a divides b”. (End of note.)

Proof. The existence of x is proved by designing a program computing x satisfying R .

Trying to establish R by means of a repetitive construct, we must choose an invariant relation. This time we apply the well-known technique of replacing a constant by a variable, and replace the constant K by the variable k . Introducing d = 2k for the sake of brevity, we then get

P:d = 2k and 1 ≤ x < d and d|(xp-r) and odd(x) .
This choice of invariant relation P is suggested by the observation that R is trivial to satisfy for K 1; hence P is trivial to establish initially. The simplest structure to try for our program is therefore:
          x, k, d :=1, 1, 2 {P};
do k ≠ K → “increase k by 1 under invariance of F” od {R} .
Increasing k by 1 (together with doubling d ) can only violate the term d|(xp-r) . The weakest precondition that d:= 2*d does not do so is —according to the axiom of assignment— (2*d)|(xp-r) . Hence an acceptable component for “increase k by 1 under invariance of P” is
(2*d)|(xp-r) → k, d := k+1, 2*d .
In the case non (2*d)l(xp-r) we conclude from d|(xp-r) that xp-r is an odd multiple of d . Because d is even, and p and x are odd, the binomial expansion tells us that (x+d)p-xp is an odd multiple of d and that hence (x+d)p-r is a multiple of 2*d . Because also d is doubled, x < d remains true under x:= x+d , because d is even odd(x) obviously remains true, and our program becomes:
  x, k, d := 1, 1, 2 {P};
do k ≠ K → if (2*d)|(xp-r) → k, d := k+1, 2*d {P}
    non (2*d)|(xp-r) → x, k, d := x+d, k+1, 2*d {P}
                    fi {P}
od {R}
Because this program obviously terminates, its existence proves the theorem. (End of proof.)

*              *
*

With the argument as given, the above program was found in five minutes. I only mention this in reply to Zohar Manna and Richard Waldinger, who wrote in “Synthesis: Dreams ⇒ Programs” (SR1 Technical Note 156, November 1977)

“Our instructors at the Structured Programming School have urged us to find the appropriate invariant assertion before introducing a loop. But how are we to select the successful invariant when there are so many promising candidates around? [...] Recursion seems to be the ideal vehicle for systematic program construction [...]. In choosing to emphasize iteration instead, the proponents of structured programming have had to resort to more dubious (sic!) means.”

Although I haven’t used the term Structured Programming any more for at least five years, and although I have a vested interest in recursion, yet I felt addressed by the two gentlemen. So it seemed only appropriate to record that the “more dubious means” have —again!— been pretty effective. (I have evidence that, despite the existence of this very simple solution, the problem is not trivial: many computing scientists could not solve the programming problem within an hour. Try it on your colleagues, if you don’t believe me.)

 

Plataanstraat 5prof.dr.Edsger W.Dijkstra
5671 AL NuenenBurroughs Research Fellow
The Netherlands

Transcribed by Martin P.M. van der Burgt
Last revision 2015-04-10 .