Integral Sequences of Infinite Length Whose Terms Are Relatively Prime

It is given in Weil and Rosenlicht ([1], p. 15) that   2 2 1, 1 1 m n c c G.C.D.    (resp. 2) for all non-negative integers m and n with if c is any even (resp. odd) integer. In the present paper we generalize this. Our purpose is to give other integral sequences m n    1 n n y   such that   G.C.D. , 1 m n y y  for all positive integers m and n with . Roughly speaking we show the following 1) and 2). 1) There are infinitely many polynomial sequences m   n n      1 n f X X    Z    , m n such that for all positive integers m and n with   1 a   G.C.D. f a f m n  and infinitely many rational integers a. 2) There are polynomial sequences     1 , n n   , g X Y    Z X Y such that       1 b G.C.D. , , m n b g a , g a  for all positive integers m and n with and arbitrary (rational or odd) integers a and b with m n    G.C.D. , 1 a b  . Main results of the present paper are Theorems 1 and 2, and Corollaries 3, 4 and 5.


Introduction
The numbers are called Fermat numbers.Fermat conjectured that F n were all prime numbers.One has , . By now, no Fermat prime has been found except for j .In Euclid's books was given the proof of existence of infinitely many prime numbers.By proving G.C.D. if , Pólya gave another proof of that, cf. ( [2], Theorem 16, p. 14) and ( [3], exercise (viii), p. 7).Weil and Rosenlicht ( [1], p. 15) considered not only 65537 for any rational integer c. e.g.[4][5][6][7][8]).Below in this paper we write in [1] asserts for all positive integers m and n with if c is even (resp.odd).
We generalize this.Let p denote any odd prime number, and let v denote any rational integer.In Theorem 2 in Section 3 below we show that m n  (resp.p) for all positive integers m and n with if v is not congruent modulo p to 1 (resp.if v is congruent modulo p to 1).Our first proof of Theorem 2 uses Elementary Number Theory.Our second proof of Theorem 2 uses Algebraic Number Theory and Theory of Cyclotomic Fields.In Corollary 5 in Section 4 we also show that for all positive inte- gers m and n with  and all rational integers v.In Corollary 4 in Section 3 we study also p q  where p and q are arbitrary odd prime numbers with .The case of for any non-negative integer u.Cf.Corollary 3 in Section 3. , In Section 2 (resp.4) we consider .
for all positive integers m and n with and all rational integers a and b with .In Theorem 3 in Section 4 we show n m (resp.2) for all positive integers m and n with and all rational integers a and b with ab (resp. ) and .The case of Theorem 3 gives a proof of Exercise IV.3 in [1].for all positive integers m and n with .
. We have also , factoring a and b into products of prime numbers, we have In Euclid's books was given the proof of the classical well known theorem that there are infinitely many prime numbers.Theorem , (see e.g.[4][5][6][7][8]).Let m and  n be arbitrary positive integers with Since there are no common roots of , , , whose leading coefficients are 1, (see e.g.[4][5][6][7][8]).Use Gauss Lemma for polynomials over the quotient ring of a factorial ring, (see e.g. ( [5], pp.181-182)).By applying it to We may put in , see e.g. [4-8].We give and Theorem 2. Let p be any odd prime number, and let v be any rational integer.Then we have the following.
Case 1 that v is not congruent modulo p to 1: p for all rational integers m and n with .Case 2 that v is congruent modulo to 1:

p T T
 for all rational integers m and n with .We give two proofs.The first one uses Elementary Number Theory.The second one uses (local and global) Algebraic Number Theory and Theory of Cyclotomic Fields for which cf. [4][5][6][7][8][9].
Proof 1.We have .Put .We have There is a rational integer We have or p.
In Case 2: We have .We have .
We give another proof of Theorem 2.

Proof 2. Let Recall
In Case 1: Now assume that there is such a prime ideal P of B that satisfies   is a primitive -th root of unity.By the theory of cyclotomic fields (cf.[4][5][6][7][8]), is a unique prime ideal  of B lying above pZ, and . We have .Hence   and .

P pZ P
We have 1 This result implies the following.If v is not congruent modulo p to 1, there is no prime ideal J with    . Let P denote the unique prime ideal in B lying above pZ, and let P B ˆ denote the localization of B at P. Let P B denote the completion of P B with respect to the P-adic (non-Archimedean) absolute value.We use local and global Algebraic Number Theory, cf.[5,9].We have   above.Therefore we get Case 2 of Theorem 2 is proven.
For each positive integer , let m denote a prime number dividing in Case 1 of Theorem 2.
 , m p v Corollary 2. We have m q  if .There are infinitely many prime numbers.
EXAMPLE of Theorem 2. Let and let be a rational integer which is not congruent modulo 5 to 1. Then we have that 1  and are relatively prime for all rational integers and with .
We give some computations.
(We used "Scientific WorkPlace", Version 5.5, MacKichan Software, 19307 8th Avenue NE, Suite C, Poulsbo, WA 98370, USA, for the computations).Corollary 3 of Theorem 2. Let be any odd prime number, and let be any rational integer.Then we have the following. Case for all rational integers and with .n Case 2 that v is congruent modulo to : for all rational integers m and n with .Proof.By ( [6], p. 280), p T   for any positive integer u.Then by Theorem 2, Corollary 3 follows.
Corollary 4 of Theorem 2. Let p and q be arbitrary odd prime numbers with , and let be any rational integer.Then we have the following.p  Case 1 that is not congruent modulo to 1: for all rational integers m and n with .Case 2 that and v p :   for all rational integers m and n with .Case 3 that and that v is not congruent modulo p to 1: We have , , , 1 or p for all rational integers m and n with .Proof.From ( [6], p. 280) we have for any positive integer u.Hence In Case 2: We have from Theorem 2. Hence it follows that In Case 3: From , the order of   divides q and .Since v is not congruent modulo p to 1, the order of is q.
Hence  

. q p
From Theorem 2, we have , , Here we use From Corollary 4 of Theorem 2 we obtain: Let and q be arbitrary odd prime numbers with  , and let be any rational integer.If p is not congruent modulo q to 1, for all rational integers m and n with .

Proof of Exercise IV.3 in [1]
Let us quote the exercise.Exercise IV. ).From .

Let m and n be arbitrary rational integers with
. Then we have: Proof.We have

Acknowledgements
The author concludes that the topic of the present paper relates to Algebraic Number Theory and Theory of Cyclotomic Fields.He would like to thank the referee for valuable suggestions for the important improvement of this paper.
Let n be any positive integer, and let n  be any primitive n-th root of unity.Let degree whose roots contain n  and whose leading coefficient is 1.One has that   n  n T does not depend on choice of  in   n We let G.C.D. denote "greatest common divisor" as usual.One has

1 :
Let   be any divisor of  .Then .
B denote the ring of the algebraic integers in any positive integer u.We have f v is even, We show first p Let p be any odd prime number, and let v be any rational integer.Then we have