Raphael M. Robinson

Undecidability and Nonperiodicity for Tilings of the Plane.

This paper is related to the work of Hao Wang and others growing out of a problem which he proposed in [8], w 4.1. Suppose that we are given a finite set of unit squares with colored edges, placed with their edges horizontal and vertical. We are interested in tiling the plane with copies of these tiles obtained by translation only. The tiles are to be placed with their vertices at lattice points, and abutting edges must have the same color. Wang raised the question whether there is a general method of deciding which finite sets of colored squares can be used to tile the plane in this way. He also discussed the relation of this problem to the decision problem for certain classes of formulas of the predicate calculus, but we shall consider only the geometrical problem here. Suppose that we have a tiling of the plane of this type which has a horizontal period. That is, we assume that the tiling remains invariant under a certain horizontal translation. There will then be a vertical strip which can be repeated to cover the plane. This strip has only a finite number of different horizontal cross sections, and hence has two which are alike. Thus the same tiles may be used to construct a tiling which has a vertical period as well as a horizontal period. A similar argument can be used even when the given period is not horizontal. That is, if a set of tiles permits a periodic tiling, then it also permits a doubly periodic tiling. In any such tiling, we can find equal horizontal and vertical periods, and hence can find a square of some size which repeats to cover the plane. Wang made the conjecture, since proved false, that any set of tiles which permits a tiling of the plane also permits a periodic tiling. He pointed out that if this conjecture were true, then we would have a decision method for an arbitrary set of tiles. Indeed, it would be sufficient to form all possible squares from the given set of tiles, starting with the smaller squares and working up, until we either reach a square which can be repeated periodically, or we find a square of a certain size which cannot be tiled at all. The latter will always happen if tiling of the whole plane

Mersenne and Fermat numbers

The first seventeen even perfect numbers are therefore obtained by substituting these values of n in the expression 2 n-(2n -1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 21271, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald [1]; see also Kraitchik [4]. The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer [7] and discussed by Uhler [13]. It is clear that 2n 1 can be factored algebraically if n is composite; hence 2n -1 cannot be prime unless n is prime. Fermats theorem yields a factor of 2n -1 only when n +1 is prime, and hence does not determine any additional cases in which 2n-1 is known to be composite. On the other hand, it follows from Eulers criterion that if n_0, 3 (mod 4) and 2n+1 is prime, then 2n+1 is a factor of 2n-1. Thus, in addition to cases in which n is composite, we see that 2n 1 is composite when 2n+1 is prime as well as n, provided that n 3 (mod 4) and n >3. Aside from this, factors of 2n -1 are known only in individual cases. If no factor is known, the best way to find out whether 2n -1 is prime is to apply a test due essentially to Lucas, but stated in a simplified form by Lehmer [6, Theorem 5.4].

MINSKY'S SMALL UNIVERSAL TURING MACHINE

Marvin L. Minsky constructed a 4-symbol 7-state universal Turing machine in 1962. It was first announced in a postscript to [2] and is also described in [3, Sec. 14.8]. This paper contains everything that is needed for an understanding of his machine, including a complete description of its operation. Minskys machine remains one of the minimal known universal Turing machines. That is, there is no known such machine which decreases one parameter without increasing the other. However, Rogozhin [6], [7] has constructed seven universal machines with the following parameters: His 4-symbol 7-state machine is somewhat different from Minskys, but all of his machines use a construction similar to that used by Minsky. The following corrections should be noted: First machine, for q600Lq1 read q600Lq7; second machine, for q411Rq4 read q411Rq10; last machine, for q2b2bLq2 read . A generalized Turing machine with 4 symbols and 7 states, closely related to Minskys, was constructed and used in [5].

Binary Relations as Primitive Notions in Elementary Geometry

Publisher Summary This chapter considers equidistance and the order of points on a line as the standard primitive notions of Euclidean, hyperbolic, or elliptic geometry. In specific, equidistance is a quaternary relation. The order of points on a line is described by the ternary relation of betweenness in Euclidean or hyperbolic geometry, and by the quaternary relation of cyclic order in elliptic geometry. the ternary relation of a point being equally distant from two other points can be used as the only primitive notion of Euclidean geometry of two or more dimensions. The chapter provides a detailed discussion of Pieris relation. The chapter also describes binary relation as a suitable primitive notion for elliptic geometry.

The Converse of Fermat's Theorem

1. Basic theorems. In this paper, we discuss some of the known results concerning the use of the converse of Fermats theorem as a test for primeness, and prove several results, believed to be new, which clarify certain points and also delimit to some extent the nature of possible improvements. The proofs of the known results are included, so that only the standard material of elementary number theory is presupposed. Fermats theorem states that if N is prime and (a, N) = 1, then

Finite sets of points on a sphere with each nearest to five others

For n = 12, the points form the vertices of an icosahedron. For n = 24, 60, the points form the vertices of Archimedean polyhedra. In the remaining two cases, n = 48, 120, the configurations do not seem to have been considered previously. A detailed description of the various configurations is given in § 7. It may be noted here that none of the configurations except the first have central symmetry; indeed, the others exist in right and left hand versions. Notice that if a is the minimum spherical distance between any pair of points, then there cannot be more than five points at the distance a from any point P of the set. Indeed, if P is joined by the shortest arcs to all the points at a distance a from it, then each angle at P is > 60 °. Thus we have assumed that each point of the set has the maximum possible number of closest possible neighbors. An assumption which appears weaker than the one made, but which is easily seen to be sufficient, is that, for each point P of the set, there are at least five points of the set which are as near to P as any point is to P. Indeed, if we start with two of the points on the sphere at the minimum possible distance a from each other, then each of these must be at the distance a from four others. Working out, we obtain a set of points satisfying the original conditions. These points will form one of the five configurations, and do not leave room for any additional points. The same problem may be considered in the elliptic plane. A configuration of n points in the elliptic plane corresponds to a configuration of 2n points on the sphere with central symmetry. Since only one of the five configurations obtained in the main theorem has central symmetry, we deduce the following corollary.

A generalization of Picard’s and related theorems

A device for the pressing of gummed labels or foils against objects which are moved along a straight or curved path, comprises one or more pressing elements disposed one behind the other and movably advanced in synchronism with the objects along a path that is convexly curved with respect to the first path. Each pressing element has a pliable pressure pad disposed at the level of the area which is to be provided with the foil and situated at varying distances from the pressing element. The pressure pad comprises a plurality of elements disposed side by side, which independently of one another exert a point or line pressure during the pressing action.

Unsymmetrical approximation of irrational numbers

1 A 1 < £ < , 5B B SB so that we have the classical theorem of Hurwitz. For other values of r, approximations from both sides are permitted, but the errors allowed on the two sides are different; hence the term unsymmetrical approximation. The result here was new, and is so related to Hurwitzs inequality that one side is strengthened and the other weakened. Notice that the result for r > l is weaker than the result for r < l . For suppose tha t r > 1, and apply the theorem with r replaced by 1/r to the irrational number — £. In this way, the permissible errors on the right and left are interchanged, and we see that £ has infinitely many approximations A/B satisfying

A Curious Trigonometric Identity

I sinh (x+ iy) I = I sinh x+ sinh iyI . We shall now show that z, sin z, and sinh z are essentially the only functions with this property.* THEOREM. If f(z) is regular for f zf <r, and satisfies the functional equation I f(x + iy) = f(x) + f(iy) for real values of x and y, then f(z) = Az, f(z) = A sin bz, or f(z) = A sinh bz, where A and b are constants, and b is real. Proof. From the preceding, it is clear that these functions do satisfy the functional equation. It remains to show that there is no other solution. Suppose that f(z) is regular for I z j <r, and satisfies the functional equation. Putting z = 0, we find that f(O) = 0. Hence we may put

On the Distribution of Certain Algebraic Integers

w 1. Introduction What point sets in the complex plane contain infinitely many sets of conjugate algebraic integers ? A basic contribution to this question was made by FEKETE [1], who showed that a bounded closed set E with transfinite diameter less than 1 can contain only a finite number of such sets of conjugates. The strict converse is not true, but a substitute is furnished by FEKETE and SZEG(J [2], Theorem K, which states that if E is symmetric to the real axis and the transfinite diameter of E is at least 1, then any open set D including E will contain infinitely many sets of conjugate algebraic integers. We may say briefly that there are infinitely many sets of conjugate algebraic integers near E. (An analogous theorem about real point sets was proved in [5].) These results may be combined in the statement that, for a bounded closed set E which is symmetric to the real axis, there are infinitely many sets of conjugate algebraic integers near E if and only if the transfinite diameter of E is at least 1. The condition that E is bounded may be dropped, if we define the terms properly. When we speak of an unbounded set E being closed, this will be understood in the spherical sense; that is, E must contain the point at infinity. Any open set D containing E will then contain all points outside of some circle, and hence will certainly contain infinitely many sets of conjugate algebraic integers. By definition, the transfinite diameter of an unbounded closed set is infinite.