On a Conjecture of Farhi

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1 47 6 Journl of Integer Sequences, Vol. 7 04, Article 4..8 On Conjecture of Frhi Soufine Mezroui, Abdelmlek Azizi, nd M hmmed Zine Lbortoire ACSA Dértement de Mthémtiques et Informtique Université Mohmmed Premier Oujd Morocco mezroui.soufine@yhoo.fr bdelmlekzizi@yhoo.fr zine00@yhoo.fr Abstrct Recently, Frhi showed tht every nturl number N mod 4 cn be written s the sum of three numbers of the form n N. He conjectured tht this result remins true even if N mod 4. In this note, we rove this sttement. n Introduction Throughout this note, we let N nd Z, resectively, denote the set of the non-negtive integers nd the set of the integers. We let nd denote the integer-rt nd the frctionl-rt functions. Let X be set. We denote the crdinlity of X by #X. We lso recll tht is the Jcobi symbol. Recently, Frhi [] showed tht every nturl number N mod 4 cn be written s the sum of three numbers of the form n N. He conjectured tht this result remins true even if N mod 4. We recll his conjecture. n Conjecture. Every nturl number cn be written s the sum of three numbers of the form n N. n In fct, he roosed more generl conjecture.

2 Conjecture. Let k be n integer. There then exists ositive integer k tht stisfies the following roerty: every nturl number cn be written s the sum of k + numbers of the form n N. n k k In this note, we rove Conjecture. Proof of Conjecture We recll Legendre s theorem [,. 9], which is necessry tool for our roof: Theorem. Every nturl number not of the form 4 h 8k+7h,k N cn be reresented s the sum of three squres of nturl numbers. We note tht since 4 h 8k +7 is congruent to 0,4 or 7 modulo 8, every nturl number not congruent to 0,4 or 7 modulo 8 cn be reresented s the sum of three squres of nturl numbers. We will use this result lter. Let r n be the number of reresenttions of the ositive integer n s the sum of three squres of integers. The following theorem rovides n interesting formul for r n, which cn be roven using the theory of modulr functions. Theorem 4 see []. For ny ositive integer n, we hve r n = 6 nχ nk π n b + b where b = b is the lrgest integer such tht b n, b n, K = m m, m= nd if 4 is the highest ower of 4 dividing n, then 0, if 4 n 7 mod 8; χ n =, if 4 n mod 8;, if 4 n,,5,6 mod 8. + We will require the following technicl lemm. Lemm 5. For ny ositive integer n mod 8, we hve r 9n > r n.

3 Proof. We hve 9n r 9n = 6 9nχ 9nK 6n π b + b 9 b n, where b = b denotes the lrgest integer for which b 9n. Since n mod8, it follows tht 4 0 = is the highest ower of 4 dividing n. This result imlies tht χ n =. Similrly, we hve 9n mod8. Thus, 4 0 = is the highest ower of 4 dividing 9n, which gives χ 9n = χ n =. Conversely, it follows from [,. 84] tht K 6n = K 4 n = K. Since n mod 8, it follows from Legendre s theorem tht n cn be reresented s the sum of three squres of nturl numbers. Thus, r n 0. Dividing through by r n then yields n identity equivlent to r 9n r n = 9n n b b b b 9 b n b n Let with n. Thus, b = b is the lrgest integer for which b n. Therefore, one obtins b = b = b = b. Furthermore, we hve 9 b n b n b n b n = = = For every with n, we then hve b + b b n r 9n r n = + b b 9 b n =. Thus, two cses re evident: if n, then b n b b b + b Otherwise, does not divide n, so r 9n r n = b b b n, 9 b n.

4 We now show tht in ll cses, r 9n > r n. If does not divide n, b = b = is imlied to be the lrgest integer for which b 9n. One obtins r 9n r n = + n. We hve =, or 4 nd so >, which gives the result r 9n > r n. If n, then b resectively b is the lrgest integer for which b n resectively b 9n. Hence, r 9n r n = b+ n b b b n b b We hve = b n + b b b + b b + b b n b n =, or 4. One obtins the following in ll cses: b n b This result imlies b b+ b n. b Conversely, result, r 9n > r n. b n > b n b. Thus, we obtin the desired Theorem 6. Every nturl number N mod 4 cn be written s the sum of three numbers of the form n N. n Proof. We my write N = +4k with k N. Thus, N + = 9+8k. We now define two sets S nd S s follows: { } S =,b,c Z : +b +c = +8k, { } S =,b,c Z : +b +c = 9+8k. 4

5 By the definition of r, we hve #S = r 9 + 8k nd #S = r + 8k. Since +8k mod8, we ly Lemm 5 to obtin r 9+8k > r +8k r +8k. One obtins r 9 + 8k > r + 8k, which is equivlent to #S > #S. We note tht this lst result is the key to the roof. Let us define the m f : S S,b,c,b,c. We see esily tht f is well defined nd injective. Since #S > #S, we cn find,b,c S such tht,b,c / fs. Furthermore, we hve + b + c = 9 + 8k 0 mod, then either b c mod or b c 0 mod. The lst cse cnnot hold becuse one of the elements,, b nd c, is not divisible by,b,c / fs. Thus, b c mod nd we hve N + = +8k = + b + c b c b c = b c Since b c mod, then + + = + + =, which gives b c N = + +. We relce,b,c Z by, b, c N to obtin the desired solution. The conjecture is roven. Acknowledgments The uthors would like to thnk the editor nd the nonymous reder for their invluble comments. This work ws suorted by Acdémie Hssn under the roject Mthémtiques et Alictions: crytogrhie nd URAC6-CNRST. References [] B. Frhi, On the reresenttion of the nturl numbers s the sum of three terms of the sequence n, J. Integer Seq. 6 0, Article.6.4. [] P. T. Btemn, On the reresenttion of number s the sum of three squres, Trns. Amer. Mth. Soc. 7 95,

6 [] A. M. Legendre, Théorie des Nombres, rd ed., Vol., Mthemtics Subject Clssifiction: Primry B. Keywords: dditive bse, Legendre s theorem, reresenttion of n integer s the sum of three squres. Received July 5 0; revised versions received July 0; November 0; December 0 0. Published in Journl of Integer Sequences, December 0 0. Revision, Jnury 04. Return to Journl of Integer Sequences home ge. 6