Amicable numbers

Two numbers are called amicable* if each equals the sum of the aliquot divisors of the other.

In the ninth century the Arab Thâbit ben Korrah¹⁰ (prop. 10) noted that $2^nht$ and $2^ns$ are amicable numbers if

$$\tag{1}h=3\cdot2^n-1,\qquad t=3\cdot2^{n-1}-1,\qquad s=9\cdot2^{2n-1}-1$$

are primes $>2$, literally, if

$$h=z+2^n,\quad t=z-2^{n-1},\quad z=1+2+\cdots+2^n,\quad s=(2^{n+1}+2^{n-2})2^{n+1}-1.$$

The term used for amicable numbers was se invicem amantes. In the article in which F. Woepcke¹⁰ translated this Arabic manuscript into French, he noted that a definition of these numbers, called congeneres, occurs in the 51st treatise (on arithmetic) of Ikhovân Alçafâ, manuscript 1105, anciens fonds arabes, p. 15, of the National Library of Paris.

Descartes³⁵⁵ gave the rule: Take (2 or) any power of 2 such that its triple less 1, its sextuple less 1, and the 18-fold of its square less 1 are all primes;* the product of the last prime by the double of the assumed power of 2 is one of a pair of amicable numbers. Starting with the powers 2, 8, 64, we get 284, 18416, 9437056, whose aliquot parts make 220, etc. Thus the third pair is

$$9363584=2^7\cdot191\cdot383,\qquad9437056=2^7\cdot73727.$$

Descartes³⁵⁶ stated that Fermat's rule agrees exactly with his own.

Amicable numbers

Two positive integers $M,N$ are amicable if $s(M)=N$ and $s(N)=M$, where $s$ is the sum of proper positive divisors. Dickson also records amiable, agreeable, befreundete, and verwandte.

Thabit’s construction

Dickson reports that Thabit ben Korrah¹⁰ gives the amicable pair $2^nht,2^nr$ when

$$h=3\cdot2^n-1,\qquad t=3\cdot2^{n-1}-1,\qquad r=9\cdot2^{2n-1}-1$$

are all primes greater than 2. In the literal formulation, put

$$z=1+2+\cdots+2^n,\quad h=z+2^n,\quad t=z-2^{n-1},\quad r=(2^{n+1}+2^{n-2})2^{n+1}-1.$$

The Latin term is se invicem amantes. In his French translation of this Arabic manuscript, Woepcke¹⁰ notes a definition under the name congeneres in the arithmetic treatise (number 51) of Ikhovân Alçafâ, manuscript 1105, anciens fonds arabes, p. 15, National Library of Paris.

Descartes’ formulation

Descartes³⁵⁵ takes a power of 2, $t\ge2$, such that $3t-1$, $6t-1$ and $18t^2-1$ are prime. Then $2t(18t^2-1)$ is one member of an amicable pair. The initial powers $2,8,64$ give $284,18416,9437056$; their proper-divisor sums give the other members, starting with 220. The third pair is

$$9363584=2^7\cdot191\cdot383,\qquad9437056=2^7\cdot73727.$$

Dickson identifies this with formula (1) on taking $t=2^{n-1}$. Descartes³⁵⁶ says that Fermat's rule agrees exactly with his own.

A common total divisor sum

Two distinct positive integers $M,N$ form an amicable pair if $s(M)=N$ and $s(N)=M$. Equivalently,

$$\sigma(M)=\sigma(N)=M+N.$$

Indeed, subtracting $M$ or $N$ from this equality gives the two defining equations. If equal entries are allowed, the condition reduces to a repeated perfect number; we separate that case from pairs of distinct numbers.

For $220=2^2\cdot5\cdot11$ and $284=2^2\cdot71$,

$$\sigma(220)=7\cdot6\cdot12=504=7\cdot72=\sigma(284),$$

and $220+284=504$. Equality of the two divisor sums alone is insufficient: their common value must also equal the sum of the two numbers.

The normalized sums satisfy a useful identity, reported by Haas on p. 48:

$$\frac1{A(M)}+\frac1{A(N)}=\frac M{M+N}+\frac N{M+N}=1.$$

If $M<N$, then $A(M)>2$ and $A(N)<2$: the smaller member is abundant and the larger is deficient. Thus equal total divisor sums coexist with different abundancy ratios.

Thabit’s conditional construction

Proposition. Let $n\ge2$. If

$$p=3\cdot2^{n-1}-1,\qquad q=3\cdot2^n-1,\qquad r=9\cdot2^{2n-1}-1$$

are all prime, then $M=2^npq$ and $N=2^nr$ are amicable.

Proof. Put $t=2^{n-1}$. Then $p=3t-1$, $q=6t-1$, $r=18t^2-1$, and $2^n=2t$. The primes are distinct and odd, so

$$\sigma(M)=(4t-1)(p+1)(q+1)=(4t-1)18t^2=\sigma(N).$$

Meanwhile

$$M+N=2t\big[(3t-1)(6t-1)+(18t^2-1)\big]=18t^2(4t-1).$$

The common-sum criterion proves the claim.

The condition $n\ge2$ matters. For $n=1$, $p=2$ would overlap the power-of-two factor, so the divisor-product calculation used in the proof would be wrong. Nor is primality automatic: for $n=3$, $r=287=7\cdot41$. The construction is not a proof of infinitely many pairs.

The same construction in different parameters

Descartes starts from the power $t$; Thabit's formula uses $n$ with $t=2^{n-1}$. This change of parameter produces exactly the same three prime tests and pair:

This is an explicit equivalence of formulas, not evidence about which historical author first knew every part of the method. Dickson's attribution and correspondence notices supply the historical account.

A broader construction needs to vary the common factor as well as the remaining primes. Euler's five problems do this. They also expose why a successful prime formula need not exhaust amicable pairs.