AN ANALOGUE IN CERTAIN UNIQUE FACTORIZATION DOMAINS OF THE EUCLIIEULER THEOREM ON PERFECT NUMBERS

We show that there exists a natural extention of the sum of divisors function to all unique factorization domains F having a finite number of units such that if a perfect number in F is defined to be an integer r/ whose proper divisors sum to % then the analogue of Euclid's theorem giving the sufficient condition that an integer be an even perfect number holds in F, and an analogue of the Euclid-Euler theorem giving the necessary and sufficient condition that an even integer be perfect holds in those domains having more than two units, i. e., in Q(/:)) and

function.The sum of divisors function, in this respect, stands apart from the other familiar number-theoretic functions.The Euler phi-function and the number of divisors function, for example, are readily extended to a unique factorization domain having a finite number of units, for they are simply counting functions, and, the Moebius function was defined in Q(/:]-) at the turn of the century (in 1901 by Gegenbaner [1] (see Dickson [2], vol. 1, p. 447)).The analogue of each of the best-known results involving these functions, including, of course, the generalized Fermat theorem and the Moebius inversion formula have been shown to hold in those fields in which these functions have been defined.
The most widely-known theorem involving the sum of divisors function, apart, possibly, from the theorem establishing the multiplicative nature of the function, is the Euclid-Euler theorem characterizing the even perfect numbers in Q: THEOREM E-E.A rational integer n is an even perfect number iff there exist rational primes p and 2 p-such that n 2P-1(2 p-1).
It is the analogue of this theorem which reason would suggest must be provable if an extension of the sum of divisors concept to an algebraic number field is to be accepted as valid and appropriate.Indeed, nearly all researchers who have examined this problem and have obtained publishable results have considered the existence of perfect numbers in the field.
That the analogue of Theorem E-E has not been proven in an algebraic extension of Q (this is not quite true--see our comments below) is related to two problems which arise.TiLe first of these is that each of the concepts "positive", "sum of divisors", "Mersenne number", "perfect number", and "even" must have a counterpart in the algebraic extension of Q which is reasonable and "natural" in some sense.A moment's reflection reveals that there may be several reasonable ways to define each of these concepts, so that many combinations of the definitions are possible.This problem was discussed by Spira [3] whose definitions in Q((i-) of "s,um of divisors" and "Mersenne number" we have used in constructing our definitions in this paper.Spira proved an analogue of Euclid's theorem stating the well-known sufficient condition that an even integer be a perfect number; using Spira's definitions, the author of this paper subsequently proved [4] an analogue of Euler's converse, subject to the restriction that the perfect numbers considered are primitive, i. e., not divisible by any other perfect number.(All perfect numbers in Q are, of course, primitive.)While these results come close to meeting the criteria that the concepts have been appropriately defined in an algebraic extension of Q, they appear to fail in one important respect.As W. D. Geyer [5] has properly mentioned in his review of Hausmann and Shapiro's article [6], "Spira has generalized the notion of a perfect number to elements of Z[i] in a certain artificial way...".A perfect number had been defined as one whose divisor sum equals the product of the prime of least norm and the number itself.
However, there is implicit in Spira's definition a relationship which makes his definition of perfect number much less artificial than might appear.We will discuss this point more fully in section 5.
The second problem encountered by the researchers who have examined the question of whether an analogue of Theorem E-E can be proved in Q(4I-), and which is, in fact, encountered in any unique factorization domain K having a finite number of units, is that Euler's proof ([7], p. 88, or see [8]) that all perfect numbers are of the form 2P-1(2 p-1) for primes p and 2 p-1, and the variation of Euler's proof (apparently due to Dickson [9]) which appears in most introductory number theory texts, do not generalize to K.This is related to the fact that the sum of the divisors of a rational integer exceeds any partial sum of its divisors, whereas in K, the sum a of the divisors of an integer may be "closer" to zero than a partial sum of its divisors.(This is under the assumption that a is a mapping from K into K.)As an example, using Spira's definition of a in Q(i-) (see our Definition 2), Ia((1+2i)2)[ 3J-7, whereas a((l+2i)2) 11 In this paper, we overcome each of these difficulties to show that there exists a very nice---and very natural--extension of the sum of divisors function and the other concepts mentioned above to all unique factorization domains having a finite number of units which yields the analogue of Euclid's theorem stating the sufficient condition that an integer be a perfect units (i.,.. in QIv/z-) and Q(qr5)).Resoluli(>r ot ll,e I.,t polletl llu'liolc(l alxwe iiiwflvcs cxanining tte roasolablc alternatives to our definitions al ,ghowit:g float they Callllol lead o , analogle of Theorctn E-E.The second problql is llitrinovic 1o obai an inequality which reduces, in well-known inequality a(pt)/p >_ (p + l)/p, for p lrinie, often '1 in lesearch on odd pcrfecl mlbcrs; this inequality is then used in obtaining l,lo th,siret 2. TIlE DEFINITIONS.
Let r/ 0 be any integer in t", a.)(l be the mi(lue unit in I" st('l) theft for l)rines k rl, r in P, r/= Hr DEFINITION 2. '[he sum a of divisors ()f ! is defined multiplicativcly by (ki+l) r a(/) II 7r.
II(1 + r + + r 1) are said to be tle proper (livios of !tle I('Is of tle prodtl('t DEFINITION 4. equals l- The integer r/ is a l>,lfe('l ttnl)er if the sln of the proper divisors of 0 It i ot necessary to define even, integer in F (see Seclion 7 for additional connents on this point,)" instead, it suffices to partition the perfc<'l ml)ers ito two classes.Let r be a prime in such that r-is a unit.We note hal 1,' complains a nique 1)rim(' in P of the form "1 + a unit" except when F Q((), i which case b()th ' and 3 + o are of this tbrm.
k, DEFINITION 5.The perfect nut)er q (llr r, I', is said to be a,n E-pcl'ecl nunber in F if r r for somei.If r r for ay i. /is said to l>e an O-pelfoct uunl>er.
DEFINITION 6.The integer a(rk-l) (r k-1)/(r-1 DEFINITION 7. The perfect number r/ is said to be primitive if there exists no perfect number a r/such that a r/.A perfect number which is not primitive is said to be imprimitive. The main results of this paper are the following analogues of the Euclid and Euler theorems characterizing the even perfect numbers.M P THEOREM 1. (Analogue of Euclid's theorem).P*, r/= rP-IM is a perfect number.P Let M P be an F-Mersenne prime.If THEOREM 2. (Analogue of Thin.E-E).Let F be Q, Q(/I), or Q(-), and / be an integer in F. The integer r/ rk-1/ is a primitive E-perfect number if and only if k is a rational prime, and # is an F-Mersenne prime in P*.If F is Q or Q(vt-), there exist no other E--,perfect numbers, i. e., all E-perfect numbers are primitive.
3. DISCUSSION OF THE CONCEPTS DEFINED IN SECTION 2.
The reader will note that if F is Q, Definitions through 6 are the familiar definitions of the corresponding concepts in Q provided E-perfect and O-perfect are read as even perfect and odd perfect, in Definition 5 (except that in addition to defining a in the usual way if r/ is a positive rational integer, we have, also, defined a for r/ negative in Definition 2).Theorem is Euclid's theorem in Q, and Theorem 2 is Theorem E-E.While these definitions are, for the most part, quite natural, alternate definitions are possible, and some have been employed with partial success in previous investigations.The rationale for our choices goes beyond the desire to extend the concepts in a natural way, however, and we shall see that alternate reasonable choices for the definitions do not lead to fully satisfactory analogues of Theorem E-E.The Set P. In Q(fT) (whose units are 1, i), P has been defined to be the first quadrant of the complex plane including the positive half of the real axis and excluding the imaginary axis, in Q(4-) (whose units are :1, 2 and-1 +2 ..v.3), p is the first sextant of the complex and plane including the positive half of the real axis and excluding the axis y qrx; all remaining fields have only the units and-1, and in these fields P is the upper half-plane including the positive but not the negative real axis.
It will be seen in Section 6 that an analogue of Euler's converse of Euclid's theorem requires that the primes r in P satisfy the condition that I]a(rk)/l] > 1.It is well-known that this inequality holds in Q, and Mitrinovic ([10], p. 140) proved that it holds for c any complex number such that Re a >_ 1; it, in fact, fails to hold for infinitely many powers of each complex number a such that Re a c < 1.Our definition of P assures that in Q(qr-) and Q(q-) this inequality holds for r E P; If F Q, Q(q-), or Q(.,f:), no choice for P such that r P : Ila(rk)/rk[I > is possible, since, in each of these fields, there exist integers both of whose associates have their real part < (suggesting the reason for the failure of the Euler converse for these fields).Definition was motivated, also, by the fact that in Q(q-), P contains two primes r such that r-is a unit--the validity of the analogue of Theorem E-E for each of the primes r demonstrates the essential nature of the conccl)t "even", as we have extended it.
An alternate choice for P which has been made in at least one investigation of perfect numbers in Q(fz]-) (Randall,[11]) involved choosing P so that the positive real axis is an axis of symmetry for P. With P so defined, however, it is not possible to prove the analogue of Theorem E-E in either Q/z]-or Q(/z): In order to show that r/ is a perfect number of the form r/= (a,k-1/?, for a unit and and ( and / prime in P, one can show that a nmst be + -1 and = a k-1.However, in Q, + -1 is a prime in P only if -1 i, implying 2k/2(cos r that = k + snk)-1; but, then, fie P only if k 0 (rood 8), and fl prime implies that, in addition, k is a rational prime, which is clearly not ssible.Similarly, in Q, + -1 is a prime in P only if -1 or (1 + )/2, and in the latter case, we have 3k/2( =cos k + sin k)-which implies, above, that k 0 (rood 12), so that, again, fl is not prime; it follows that if r (3 + )/2, Q() has no E-perfect num@rs of the familiar form specified in Theorem 2. ndall showed, for this choice of P, that if a rfect humor in Q() is defin by a(y) 2, then all perft numbers in Q() have at let three prime factors and 5 is the only perft number < 106.
The Sum of Divisors Function.Previous investigations of problems involving the sum of divisors function in Q(I-) have used the concept as defined in Definition 2 [3], [4], [11] (although not all have required that the prime factors of y be in the set P as we have defined it), or, have defined a to be the sum of the norms of the divisors of r/[12].In related work, Hausmann and Shapiro [6] investigated perfect ideals over the Gaussian integers, defining the sum of divisors of an ideal to be the sum of the norms of the ideal's divisors; in this paper, the authors found only two perfect ideals and showed that there are no others having fewer than five distinct prime factors.
Perfect Number.Definition 4 is, of course, the classical definition of a perfect number.In view k.
Even Perfect Number.Our decision to classify the perfect numbers as E-perfect numbers or O-perfect numbers as opposed to defining an integer r/in F as even iff r r] is related to the fact that the laws of parity do not necessarily hold in F under this definition.They do hold if F Q(f3-); however, it is not difficult to construct examples to show that if F Q(/) and r (3 + /:5)/2, the sum of two odd integers may be odd.There is, on the other hand, no requirement that the partitioning of perfect numbers into two classes satisfies the laws of parity--the sum of two even perfect numbers in Q, for example, is not an even perfect number.
The concept of "evenness" in F may be defined by any one of the following: n is even if n is even if n is divisible by the prime factor of 2 of least norm; n is even if n is divisible by the prime in F of least norm; n is even if n is divisible by a prime of the form 1 + u where u is a unit.
W.L. MCDAN EL While these definitions ate equivalent in Q, it is toa,lily seen tiat lwy ae ot equivalent in, for exanple, Q(/i) and Q(/s?).We have not 'ho.(,nIo extond the co)<'('l)t ()f "erectness" by sing any of the first three definitions because, in each ('a.c, the analoguo fails in one or more fields F (it fails in Q(/-) for (i), atd in Q(/s) fo (ii) ,)nd (iii), tot instan('c).The concept even integer in an algebraw numb() field vill be Section 7.
F-Mersenne Number.The definition is identical to t))e definition of Morse,me numb('t in Q. in Q, M k is a prime in F only if k is a rational prime 4. TIIE PROOF OF TIIEOREM 1 AND A LOWEIt BOUNI) FOR Ila(rt)/.tll.
Theorem i readily established.
In order to prove Theorem 2, we first obtain an inequality wlich reduces to the well--known lower bound often used iu research on odd perfect numbers: a(pt)/p >_ (p + 1)/p, where equality holds iff 1.
COROLLARY.If r is a prime in Q(J:ff) and x + yf: is the associate in P of r, a()/r t > (I] rll + 2x 1)/]] rll; PROOF.Since [[r[[ I[x + Yqt:$[I and [[a(rt)[I [[1 + + (x + yq-)t[I, the corollary is merely a restatement of the theorem for z r a prime in Q(J), provided we show that x >_ 5/4.
Because x + y4-is an integer in Q(f:), x + y4-(a + b/:3-)/2, where a and b are rational integers of the same parity.Since this integer is in P, b < a, and it is therefore immediate that x= a]2isnot 112 nor 1. So, x_> 3]2 > 5]4.
If z p, a rational prime, the second inequality of Theorem 3 (or of the Corollary) becomes the bound a(pt)/p t _> (p + 1)/p.
We have stated, in the Introduction, that while Spira defined an integer r/ in Q(J:]-) to be a perfect number if a(r/)= rot t, where v o is the prime in e of least norm (that is, v o 1 + i), this definition, in fact, implies a more fundamental relationship.We shall show that in this section, and prove Theorem 2 for F Q(i).
Our Definitions 1, 2 and 6 (for F Q(Jx]')) are those used by Spira in defining P, "sum of divisors" and "Mersenne number" in Q(q]), and Spira defined the integer r/ to be even if (1 + i)l r/.Incorporating Spira's proof of the sufficiency in a theorem characterizing the even perfect numbers in Q(]), under the above definitions, the author proved [4] THEOREM A. r/ is a.n even primitive perfect number iff there exists a rational prime p (mod 8) and a Mersenne prime Mp such that r/= (1 + i)P-lMp.
We shall show that Theorem A, under Spira's definitions of even integer and perfect number, is equivalent to our analogue of Theorem E-E, i. e., to Theorem 2 with F Q(4":-]-).
Our definition of P and r in equal to rP-IM Since P Q(-i-) imply that r + r o, so, iu Theorem A, 11 is [2P/2(cos r 7r Mp (r p-1)/(r-1) p + sin p)-1]/i, it is readily seen that the condition that p (mod 8) is equivalent to Mp e P*.Finally, under Spira's definitions, rl erP-ll , for a unit and/t the product of prime factors in P, is an even perfect number iff a(r/) (1 + i)r/, and, by the observation in Section 3 concerning our Definition 4, r/is E-perfect iff a(r/) (1 +e-1)r/.It is now clear that Theorem A and Theorem 2 with F Q(:]-) are equivalent if =-i.But this is indeed the'case, for, upon observing that the associate in P of Mp is Mp iMp, we see that the perfect number r/ (1 / i)P-lMp, (M'p in Theorem A, can be written r/---irP-lMp P)" therefore r/ is of the form r/= erP-1/z, with =-i.

E-PERFECT NUMBERS IN Q(/-).
In this section, we shall complete the proof of Theorem 2.
The Form of E-Perfect Numbers in Q(I).
N k -IIMKI I.
In the proof of the following lemma, we let LEMMA 1. If, for (r,#) and k > 1, rk-1/ is a perfect number in Q(4-), then the Q(/zff)-Mersenne number M k a(rk-l) is prime.
PROOF.Assume rk-l is a perfect number in Q(-ff).Let r x + y/L-be any prime factor in P of M k.Since (r,Mk) 1, ry a(r/)= a(rk-1)a(/)= Uka(/ (6.1) implies that rlr/, and hence, /; we let a be the largest rational integer such that ra[p.Since, by the Corollary to Theorem 1, Ila(at)/atl[ > for any prime power at, we have, upon using the multiplicative property of a, [[a(r/)/rr/[[ >_ [[a(rk-1)a(ra)/rkral[ > Nk ([[r[[ + 2x-1)/llrkll.llr[IIn the proof of the Corollary to Theorem 3, we observed, for r x + y4J, that x >_ 3/2.Since the only prime in P having its real part equal to 3/2 is r, x _> 2; hence, applying the Corollary to Theorem 3 to the above inequality, ((;.2) Substituting and simplifying, we find that, fi)r each value of r, the remlting inequality doos ot hold for k > 1.Since M k is a factor of rt, it follows that II,tl > N k ]]Mk]] /2; hen('c, f()r some unit , r Mk, implying that M k is prilne.
It is clear that (6.4) holds only if cos p is positive.That is, only if p 0, ,1, or *2 (rood 12).
Since p is a rational prime, p is not congruent to 0 or i2, modulo 12, unless p 2. If p 2, however, M p r + + -, aud Ila(u)/r,ll >_ II'(r)tr(M2t)/r 2" M2tll > Ila(r)a(M2)ll/9-llM2ll Ila(M2)ll/9-13/9.Thus, p tl (rood 12).Suppose p -1 (rood 12).q'hen Mp (3 (p+l)/2 2 3P/2i)/2(r-1), and the a.ssociate in P of M is P r + a unit in Q(-42)), and under these alternate definitions, Theorem is siml)ly not true for F-Q(.,/).This exception points tip, again, the observation that the proof ia Q of Theorem E-E relies, in an essential way, upon the fact that an even integer has the property that it is divisible not by the "least" prime, but rather, by a prime of the form + u, where u is a unit.
Because many of the results concerning the form or existence of odd perfect numbers in Q are based on the inequality which we have extended to F in Theorem 3, it is anticipated that the analogues of these theorems can be obtained in Q(vi-) and Q(.,-) using the concepts of this paper.