000RM.dvi

21.2 Somos sequences 603

21.2 Somos sequences .................

Define two sequences(an)and(bn)with initial values

a 0 =a 1 =1,b 0 =0,b 1 =1,

and forn≥ 2 recursively by

an=an− 2 bn− 1 ,

bn=an− 1 +an.

Here are the first few terms

n 01234 5 6 7 8 9 ···

an 111231039490 20631 10349290 ···

bn 012351349529 21121 10369921 ···

Theorem 21.1.The terms of the sequence(bn),n ≥ 1 are relatively
prime.

Proof.If we calculate the first few terms

a 2 =a 0 b 1 =b 1 ,

a 3 =a 1 b 2 =b 2 ,

a 4 =a 2 b 3 =b 1 b 3 ,

a 5 =a 3 b 4 =b 2 b 4 ,

a 6 =a 4 b 5 =b 1 b 3 b 5 ,

a 7 =a 5 b 6 =b 2 b 4 b 6 ,

..

.

we see a pattern, namely,

a 2 n=b 1 b 3 ···b 2 n− 1 ,

a 2 n+1=b 2 b 4 ···b 2 n.

Suppose, inductively, thatb 1 ,b 2 , ...,b 2 n− 1 are relatively prime. Then,

b 2 n=a 2 n− 1 +a 2 n=b 2 b 4 ···b 2 n− 2 +b 1 b 3 ···b 2 n− 1

does not contain any divisor ofb 1 ,b 2 , ...,b 2 n− 1 ; nor does

b 2 n+1=a 2 n+a 2 n+1=b 1 b 3 ···b 2 n− 1 +b 2 b 4 ···b 2 n

contain any divisor ofb 1 ,b 2 , ...,b 2 n. It follows by induction that no two
of the termsb 1 ,b 2 , ...,bn,... contain a common divisor.