Odd Square Research Articles

On the Size of a Triple Blocking Set in PG(2, q)

Lower bounds are obtained for the size of a triple blocking set in the Desarguesian projective plane PG(2, q). These bounds are best possible for q<11and in the case q>121is an odd square.

Extending the Pythagorean Theorem to Other Geometries

Many theorems of Euclidean plane geometry can be generalized to forms that also hold for some other plane geometries. This is done by generalizing the definitions for distance and angle. To do this for the Pythagorean Theorem it suffices to generalize distance, and to do this, the odd squazreftinction 2 and its inverse, the odd square root function 1/2, are defined for x real by (note-umlauts usually signify an odd pronounciation for a vowel)

Ádám's conjecture is true in the square-free case

Ádám's conjecture [1] formulates necessary and sufficient conditions for cyclic (circulant) graphs to be isomorphic. It is known to be true if the number n of vertices is either prime ([4]), a product of two primes ([12]) or satisfies the condition n, φ( n)) = 1, where φ is Euler's function ([15]). On the other hand, it is also known that the conjecture fails if n is divisible by 8 or by an odd square. It was newly conjectured in [15] that Ádám's conjecture is true for all other values of n. We prove that the conjecture is valid whenever n is a square-free number.

On the existence of translation nets

The existence of a translation net of order s and degree r with translation group G is equivalent to the existence of r mutually disjoint subgroups of G of order s. In this paper we consider p-groups G of odd square order p 2 n and improve the known general upper bound on the number of mutually disjoint subgroups of order p n in G provided that G is not elementary abelian. This solves problem 8.2.14 in (D. Jungnickel, Latin squares, their geometries and their groups. A survey, in “Coding Theory and Design Theory II” ( D. K. Ray-Chaudhuri, Ed.), pp. 166–225, Springer, Berlin/New York, 1990.) We determine all groups of order p 4 which are translation groups of translation nets with at least three parallel classes for all prime numbers p. Furthermore, we construct ( p 3, p 2 + 1)-translation nets with non-abelian translation group of order p 6 for all odd prime numbers p.

A 35-set of type (2, 5) in PG(2, 9)

A 35-set of type (2, 5) is constructed in the Desarguesian plane of order nine, which is the first example of a set of type ( m, q + m) and ( m + q)( q 2 − q + 1) points, with m = 2, in an odd square order plane.

Re: Soviet river diversions

The paper on ‘Soviet River Diversions’ by Phil Micklin (Eos, 62(19), May 12, 1981) has just come to hand.Referring to the map on page 489, I was interested to see the estimates of river flows for the Amu and Syr Darya, which clearly show the effect of irrigation on inflows to the Aral Sea. Recently, I was passing over the northeast corner of the sea on a flight from Tashkent to Moscow when I got the impression that increasing irrigation development on the Syr Darya is likely to decrease the annual inflow even more than in the recent past. The same state of affairs has been going on in the Caspian Sea for years, as a result of irrigation development on the Volga. My impression was that the Aral Sea had shrunk considerably from the 26,000 odd square miles (67,304 km2) area quoted (from memory) in Encyclopaedia Britannica (edition circa 1970).

Polarities and ovals in the Hughes plane

SummaryIn 1946 Baer (Polarities infinite projective planes, Bull. Am. Math. Soc. 52, 77–93) showed that the absolute points of a polarity in a finite projective plane of odd non-square order always form an oval, that is, in a plane of order n there are exactly n+ 1 absolute points and no three are collinear. It is well known that the absolute points of polarities in planes of odd square order form ovals in some cases.If the oval is a subset of the set of absolute points, then the oval itself determines the polarity, and this makes it appear unlikely that the oval could be a proper subset. Among other results in the paper it is to be proved that in the regular Hughes plane there is a polarity which is determined by an oval which is a relatively small subset of the set of absolute points. Explicitly, if Ω is the Hughes plane of order q2 and A is the central subplane of order q, then every conic in Δ can be extended to an oval in Ω, and this oval determines a polarity in which there are ½(q3–q) additional absolute points.

Finite Groups the Centralizers of whose involutions have Normal 2-Complements

In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity, we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition.As is well-known, the automorphism group G = PΓL(2, q) of H= PSL(2, q), q= pn, is of the form G = LF, where L = PGL(2, q), L ⨞ G, F is cyclic of order n, L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q) acts; cf. (3, Lemma 2.1) or (7, Lemma 3.3). It follows at once (4, Lemma 2.1; 8, Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.

On the Divisors of the Discriminant of the Period Equation

Volume 3 of this Journal Sylvester [1] wrote the following postscript to his paper: In a future communication I will prove very simply that if a prime number p = ef + 1, and e itself is a prime such that e -1 contains no odd square number, then every divisor, withoue exception (other than p), of the function whose roots are the periods of the primitive pth roots of unity, must be an eth power residue. If (e -1) contains any square number the proof still holds good, except as regards factors of such square, and there is no reason at present for supposing that the theorem may not be extended to the case of these excepted factors. A year later in Comptes Rendus [2] he returns to the problem of possible exceptional divisors (i. e. divisors which are not e-th power residues), states that they divide the discriminant, shows that they do not exist for cubic equations, complains (quite rightly) that the problem is not treated in Bachmann [3] and reserves it once more for future consideration. We have not been able to find any further reference to the problem in the works of Sylvester or elsewhere. It is completely ignored by the Smith or Hilbert Reports on the Theory of Numbers as well as the National Research Council Report on Algebraic Numbers [4]. All the above mentioned sources discuss at length Kummer's work on the period equation without pointing out that Kummer [5] had essentially a proof of the nonexistence of Sylvester's exceptional divisors some thirty-five years before Sylvester proposed the problem. this paper we will give a proof proceeding along the lines outlined in Kummer [5] of the non-existence of such exceptional divisors, but we are at a loss to explain why Sylvester had to exclude divisors of e-1 which appear to the second power. His unpublished proof must have proceeded along quite different lines. More precisely we will prove:

Note on sums of four and six squares

which is equivalent to g2'(u-o-a(2u)/a_4(U), can be used to prove various results involving partition functions. Dobbie [3] recently constructed simple direct proofs of (1) and (2) that require no knowledge of elliptic functions; incidentally (2) can be derived from (1) by dividing by x -y and then letting y-*x. The writer [2] showed that by means of (2) one can give a very concise proof of the familiar formula for the number of representations of an integer as a sum of eight squares or of eight odd squares. In the present note we obtain the formulas for four and six squares in a similar manner (see for example [6, p. 307]).

On Residue Difference Sets

In recent years the subject of difference sets has attracted a considerable amount of attention in connection with problems in finite geometries [4]. Difference sets arising from higher power residues were first discussed by Chowla [1], who proved that biquadratic residues modulo p form a difference set if (p — l )/4 is an odd square. In this paper we shall prove a similar result for octic residues and develop some necessary conditions which will eliminate all odd power residue difference sets and many others. We also prove that a perfect residue difference set (that is, one in which every difference appears exactly once) contains all the powers of 2 modulo p.

Southern Patagonia: As Portrayed in Recent Literature

HIRTY years ago thousands of people who had rushed to the southern extremity of South America in search of gold turned to sheep raising as a more certain means of making a living. Today we perceive a tendency toward reversal of this trend. Not that new gold deposits have been found in this region-actually we know little more about them-but the world itself has changeda world that finds itself today with plenty of sheep and a shortage of the gold necessary to distribute those sheep, a world that needs gold far more than it needs mutton or wool or any other commodity. Indications are not wanting that this forgotten region is upon the threshold of return to public interest. Of all the regions of the earth's temperate zones it is doubtful whether there remains a single one of which man's knowledge is so incomplete as of that of the southern Patagonian Andes from the forty-fifth parallel of latitude to the outermost islands of the Tierra del Fuego archipelago. Yet we can hardly call this land of mystery remote: not one of its I8o,ooo odd square miles of mainland and islands is 200 miles distant from one of the New World's oldest trade routes. Nor has there been a dearth of interest in it. Between Magellan's discovery of the strait in I520 and the visit of the Beagle in I832 no fewer than 81 expeditions representing many nations have explored there. Between the voyage of the Beagle and the publication of Charles Darwin's Diary of the Voyage of H. M. S. 'Beagle' last year' scores of books and articles in various languages have appeared. Yet one needs only to glance over the most recent literature to realize how little is actually known, and even the latest maps disclose vast areas marked inexplorado. But, while the region has fallen into further economic obscurity with the replacement of the Strait of Magellan as a maritime channel by the Panama Canal, a new impetus seems to have been given to scientific exploration. To mention one event of importance: the most extensive of Patagonia's great inland ice fields was traversed for the first time by De Agostini in I93Itruly a noteworthy achievement. We may well begin our discussion with the problem of the present glaciation of Patagonia.

III. On some remarkable relations which obtain among the roots of the four squares into which a number may be divided, as compared with the corresponding roots of certain other numbers

The property of numbers, which is the subject of this paper, first presented itself to my attention in the case of the odd squares 1, 9, 25, 49, &amp;c. (2 n ∓ 1) 2 ; any two adjoining odd squares may be divided (each of them) into 4 square numbers, whose roots will have this remarkable relation to each other: two of them will be identically the same; and as to the other two, one of them will be 2 less, and the other will be 2 more than the roots of the preceding or subsequent odd square; for example, 25 and 49 may be divided into squares, the roots of which being placed below, will appear thus: — 25 49 so 49 81 -2, 1, 4, 2 -4, 1, 4,4 0, 2, 3, 6 -2, 2, 3, 8 or thus 0, 0, 3, 4 -2, 0, 3, 6. In comparing the roots of the adjoining odd squares, 2 roots (placed in the middle) are the same; of the others, one is 2 more, the other 2 less than the corresponding roots of the other.

A proof (by means of a series) that every number is composed of 4 square numbers, or less, without reference to the properties of prime numbers

The paper contains a proof, that if every number of the form 8 n + 4 is composed of 4 odd squares, then every number whatever must be composed of 4 square numbers or less; also a proof of the converse of this, viz. that if every number is composed of 4 square numbers or less, then every number of the form 8 n + 4 must be composed of 4 odd squares. It is then proposed to show that every number of the form 8 n + 4 is composed of 4 odd squares, by taking a number of the form 8 n + 4, viz. an odd square +3, and showing that 8 n + 4 in that case is divisible into 4 odd squares (other than the odd square and 1, 1, 1); thus 16 n 2 ± 8 n + 1 is a form that includes every odd square, and 16 n 2 ± 8 n + 4 is divisible into 4 n 2 ± 4 n + 1, 4 n 2 ± 4 n + 1, 4 n 2 ± 4 n + 1, 4 n 2 ∓ 4 n + 1.

1. On the extension of the principle of Fermat’s theorem of the polygonal numbers to the higher orders of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders

The object of this paper professes to be to ascertain whether the principle of Fermat’s theorem of the polygonal numbers may not be extended to all orders of series whose ultimate differences are con­stant. The polygonal numbers are all of the quadratic form, and they have (according to Fermat’s theorem) this property, that every number is the sum of not exceeding, 3 terms of the triangular num­bers, 4 of the square numbers, 5 of the pentagonal numbers, &amp;c. It is stated in this paper that the series of the odd squares 1,9,25,49, &amp;c. has a similar property, and that every number is the sum of not exceeding 10 odd squares. It is also stated, that a series con­sisting of the 1st and every succeeding 3rd term of the triangular series, viz. 1,10,28,35, &amp;c., has a similar property; and that every number is the sum of not exceeding 11 terms of this last series, and that this may be easily proved [it was proved in a former paper by the same author]. The term “Notation-limit” is applied to the num­ber which denotes the largest number of terms of a series necessary to express any number; and the writer states that 5,7,9,13,21 are respectively the notation-limits of the tetrahedral numbers, the octa­hedral, the cubical, the eicosahedral and the dodecahedral numbers; that 19 is the notation-limit of the series of the 4th powers; that 11 is the notation-limit of the series of the triangular numbers squared, viz. 1,9,36,100, &amp;c., and 31 the notation-limit of the series 1,28,153, &amp;c. (the sum of the odd cubes), whose general expression is 2 n 4 — n 2 .