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DISTRIBUTION OF QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI Abstra ct. In this article, shall study using elemen tary and com binatorial metho ds, the distribution of quadratic non-residues whic are not primitiv ro ots mo dulo or for an dd prime and is an in teger. 1. Intr oduction Distribution of quadratic residues, non-residues and primitiv ro ots mo dulo for an ositiv in teger is one of the classical problems in Num er Theory In this article, applying elemen tary and com binatorial metho ds, shall study the distribution of quadratic non-residues whic are not primitiv ro ots mo dulo dd prime ers. Let an ositiv in teger and an dd prime um er. denote the additiv cyclic group of order The ultiplicativ group mo dulo is denoted of order the Euler phi function. Denition 1.1. primitive ot mo dulo is generator of whenev er is cyclic. ell-kno wn result of C. F. Gauss sa ys that has primitiv ro ot if and only if or or for an ositiv in teger Moreo er, the um er of primitiv ro ots mo dulo these 's is equal to )) Denition 1.2. Let and in tegers suc that a; If the quadratic congruence (mo has an in teger solution x; then is called quadr atic esidue mo dulo Otherwise, is called quadr atic non-r esidue mo dulo n: Whenev er is cyclic and is primitiv ro ot mo dulo n; then for are all the quadratic non-residue mo dulo and for are all the quadratic residue mo dulo n: Also, for all suc that (2 )) are all the quadratic non-residues whic are not primitiv ro ots mo dulo n: or ositiv in teger n; set is primitiv ro ot mo dulo 2000 Mathematics Subje ct Classic ation. 11N69. Key wor ds and phr ases. quadratic non-residues, primitiv ro ots.

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI and is quadratic non-residue mo dulo Note that (1) (1) (2) and (2) When kno that and whenev er or or ha Also, it can easily seen that if then shall denote quadratic non-residue whic is not primitiv ro ot mo dulo QNRNP mo dulo n: Therefore, an is QNRNP mo dulo n: Recen tly zek and Somer [2 pro ed that i is either ermat prime (primes of the form 1) or or wice ermat prime. Moreo er, they pro ed that for if and only if or 18 or either or n= is equal to prime p; where 1) is also an dd prime. They also pro ed that when then In this article, shall pro the follo wing theorems. Theorem 1.1. et and any ositive inte gers. et or for any dd prime p: Then if and only if is either (i) or whenever +1 with is also prime or (ii) or whenever +1 is ermat prime. In this ase, the set is nothing but the set of al gener ators of the unique cyclic sub gr oup of or der +1 of When is not ermat prime, then it is clear from the ab discussion that := 1) When the natural question is that whether do es there exist an consecutiv pair of QNRNP mo dulo rom Theorem 1.1, kno that for all primes where is also prime um er. Theorem 1.2. et prime such that wher is also prime. Then ther do es not exist air of onse cutive QNRNP mo dulo p: In con trast to Theorem 1.2, shall pro the follo wing. Theorem 1.3. et any dd prime such that 1) Then ther exists air of onse cutive QNRNP mo dulo p: In the follo wing theorem, shall address eak er question than Theorem 1.3; but orks for arbitrary length Theorem 1.4. et any dd inte ger and inte gers. Then ther exists ositive inte ger dep ending only on and such that for every prime and (mo we have an arithmetic pr gr ession of length whose terms ar QNRNP mo dulo n; wher or Mor over, we an cho ose the ommon dier enc to QNRNP mo dulo n; whenever

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QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS 2. Preliminaries In this section, shall pro some preliminary lemmas whic will useful for pro ving our theorems. Prop osition 2.1. et any ositive inte ger and let or for any dd prime p: Then any inte ger is primitive ot mo dulo if and only if =q 6 (mo for every prime divisor of Pr of. Pro of is straigh forw ard and omit the pro of. The follo wing prop osition giv es criterion for QNRNP mo dulo whenev er or Prop osition 2.2. et any ositive inte ger. et any ositive inte ger of the form or wher is an dd prime. Then an inte ger is QNRNP mo dulo if and only if for some dd divisor of we have, (mo Pr of. Supp ose is QNRNP mo dulo n: Then, (mo If is ermat prime or wice ermat prime, then kno that ev ery non- residue is primitiv ro ot mo dulo n: Therefore, the assumption, is not suc um er. Th us there exists an dd in teger whic divides Since is not primitiv ro ot mo dulo n; Prop osition 2.1, there exists an dd prime dividing satisfying =q (mo Therefore, taking the square-ro ot of =q mo dulo n; see that (mo If (mo then taking the -th er oth the sides, it follo ws that is quadratic residue mo dulo p; con tradiction. Hence, get (mo or the con erse, let an in teger satisfying (mo (1) where is an dd divisor of Then squaring oth the sides of (1), conclude Prop osition 2.1 that cannot primitiv ro ot mo dulo n: By taking the -th er oth sides of (1), see that the righ hand side of the congruence is still as is dd and hence conclude that is quadratic non-residue mo dulo n: Th us the prop osition follo ws. Corollary 2.2.1. et prime. Supp ose is not ermat prime and divides If is QNRNP mo dulo p; then 1) is squar e-r ot of mo dulo for some dd divisor of

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI Pr of. By Lemma 2.2, it follo ws that there exists an dd divisor of suc that 1) (mo Since divides it is clear that 1) (mo and hence the result. Lemma 2.3. (K zek and Somer, [2 ]) et an dd ositive inte ger. Then (2 and (2 Theorem 2.4 (Brauer, [1]) et and ositive inte gers. Then ther exists ositive inte ger dep ending only on and such that for any artition of the set into -classes, we have ositive inte gers a; d; 1) and sd lie in only one of the 's. Using Theorem 2.4, Brauer [1] pro ed that for all large enough primes p; one can nd arbitrary long sequence of consecutiv quadratic residues (resp. non-residues) mo dulo p: Also, in series of pap ers, E. egh [3], [4 ], [5], [6 and [7] studied the distribution of primitiv ro ots mo dulo or 3. Pr oof of Theorem 1.1 Lemma 3.1. et and any ositive inte gers. et or for any dd prime p: Then if and only if is either (i) or whenever +1 with is also prime or (ii) or whenever +1 is ermat prime. Pr of. In the view of Lemma 2.3, it is enough to assume that Let where `; are ositiv in tegers suc that Case (i) 1) In this case, ha e, 1) and Hence, ould imply and Since the ositiv in teger satises ust prime um er. Therefore, those primes satises the yp othesis are of the form +1 where is also prime um er. Case (ii) 2) In this case, ha e, )) 1)) 1) 1) 1) No w, 1)

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QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS Therefore, implies and hence get, and Th us ha e, +1 whic ould imply cannot greater than If then ha +1 That is, if then the only in tegers satises the yp othesis are where is ermat prime. Con erse is trivial to establish. Pr of of The or em 1.1. Giv en that By Lemma 3.1, ha cases. Case (i) or where +1 where is also prime) Let an arbitrary elemen t. Then is quadratic non-residue mo dulo but not primitiv ro ot mo dulo n: Therefore Prop osition 2.2, kno that there exists an dd divisor of satises (mo Since +1 where is the only dd divisor of ust ha Therefore, (mo (mo +1 (mo Let the unique cyclic subgroup of Then with order of is +1 Hence as is arbitrary is the set of all generators of Case (ii) or where +1 is prime and is er of 2) Let Then Prop osition 2.2, kno that there exists an dd divisor of satisfying 1) (mo and hence +1 (mo Th us, where is the unique subgroup of of order +1 4. Pr oof of Theorem 1.2 Lemma 4.1. et prime such that wher is also prime. If a; 1) is air of QNRNP mo dulo p; then (mo Pr of. Giv en that and are QNRNP mo dulo p: Therefore, Prop osition 2.2, ha (mo and 1) 1) (mo That is, 1) (mo Hence the result. Pr of of The or em 1.2. By Lemma 3.1, kno that for these primes, there are exactly QNRNP mo dulo p: Supp ose assume that these QNRNP mo dulo are consecutiv pair, sa a; 1) Then Lemma 4.1, get, (mo end the pro of, shall, indeed, sho that is primitiv ro ot mo dulo

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI and arriv at con tradiction. pro is primitiv ro ot, ha to pro that the order of in is Since the order of is and the order of is equal to the order of it is enough to pro that is primitiv ro ot mo dulo p: By Prop osition 2.1, ha to pro that 6 (mo for ev ery prime divisor of In this case, ha and If then 1) =q and so 16 6 (mo as Hence, it is enough to pro that 6 (mo Indeed, the quadratic recipro cit la w, kno (mo and hence the theorem. 5. Pr oof of Theorem 1.3 Lemma 5.1. et prime such that et denote the total numb er of QNRNP mo dulo Then exactly 1) numb er of QNRNP mo dulo ar fol lowe by quadr atic non-r esidue mo dulo whenever wher is an dd inte ger; Otherwise, exactly half of QNRNP mo dulo is fol lowe by quadr atic non-r esidue mo dulo p: Pr of. First note that 1) 1) is dd if and only if 1) is dd if and only if where is an dd in teger. Let QNRNP mo dulo p: Let xed primitiv ro ot mo dulo p: Then there exists an dd in teger satisfying `; 1) and Therefore, is also QNRNP mo dulo p: Then ha e, (1 (mo This implies is quadratic residue mo dulo if and only if is quadratic non-residue mo dulo p: Therefore, to complete the pro of of this lemma, it is enough to sho that if is QNRNP mo dulo and 6 (mo then 6 (mo Supp ose not, that is, (mo Then, (mo Since it is clear that whic ould imply and therefore get (mo con tradiction and hence 6 (mo Similarly ha 6 (mo Note that (mo if and only if (mo 1) whic ould imply 1) as Since is dd, this happ ens precisely when where is an dd in teger. Hence the lemma. Pr of of The or em 1.3. Let an prime suc that 1) 1) If ossible, shall assume that there is no pair of consecutiv QNRNP mo dulo p: Let 1) Therefore, clearly By Lemma 5.1, kno that exactly half of QNRNP mo dulo follo ed quadratic non-residue mo dulo p: This implies, 1) um er of QNRNP mo dulo follo ed primitiv ro ots mo dulo p: Since there there are utmost 1) primitiv ro ots ailable, it follo ws that there exists QNRNP mo dulo follo ed QNRNP mo dulo p:

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QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS 6. Pr oof of Theorem 1.4 Giv en that is an dd in teger and is an in teger. Put and in Theorem 2.4. get natural um er dep ending only on and suc that for an -partitioning of the set ha ositiv in tegers a; d; d; 1) and whic are less than or equal to and lying in exactly one of the classes. Cho ose prime suc that (mo By Diric hlet's prime um er theorem on arithmetic progression, suc prime exists and there are innitely man suc primes. Let xed primitiv ro ot mo dulo Note that for eac there exists unique in teger 1) satisfying (mo partition the set in to parts as follo ws. if and only if (mo Since there exists an arithmetic progression of length sa a; d; 1) together with its common dierence lying in for some By the denition of our partition, ha id (mo and (mo where for eac satisfying (mo Since 's run through single residue class mo dulo can as ell assume, if necessary suitable translation, that (mo No w, ho ose an in teger suc that (mo 2) and (mo Then see that (mo Since is an dd in teger and are ev en in tegers, get, are dd in tegers together with (mo Therefore, divides the gcd( ; 1) Putting (mo get, a; d; 1) d; are QNRNP If is an dd in teger, then is also primitiv ro ot mo dulo If is an ev en in teger, then put whic is an dd in teger and hence it is primitiv ro ot mo dulo No the pro of is similar to case when and lea it to the readers. Before conclude this section, shall raise the follo wing questions. (1) Can Theorems 1.4 true for all large enough primes (2) What is the general prop ert of the set of all ositiv in tegers satisfying for an giv en ositiv in teger

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI Ac kno wledgmen t. are thankful to Professor M. zek for sending his pap er [2 ]. Also, are grateful to Professor D. Rohrlic oin ting out an error in the previous ersion of this pap er. References [1] A. Brauer, Ub er Sequenzen on otenzresten, Sitzungsb erichte der Pr eubischen kademie der Wissenschaften (1928), 9-16. [2] M. zek, L. Somer, necessary and sucien condition for the primalit of ermat um ers, Math. Bohem., 126 (2001), no. 3, 541-549. [3] E. egh, airs of consecutiv primitiv ro ots mo dulo prime, Pr c. mer. Math. So c. 19 (1968), 1169-1170. [4] E. egh, Primitiv ro ots mo dulo prime as consecutiv terms of an arithmetic progression, J. eine ngew. Math. 235 (1969), 185-188. [5] E. egh, Primitiv ro ots mo dulo prime as consecutiv terms of an arithmetic progression I, J. eine ngew. Math. 244 (1970), 108-111. [6] E. egh, note on the distribution of the primitiv ro ots of prime, J. Numb er The ory (1971), 13-18. [7] E. egh, Primitiv ro ots mo dulo prime as consecutiv terms of an arithmetic progression I, J. eine ngew. Math. 256 (1972), 130-137. School of Ma thema tics, Harish Chandra Resear ch Institute, Chha tna ad, Jhusi, Allahabad 211019, India. E-mail addr ess S. Gun: sanoli@mri.ernet.in E-mail addr ess B. Ramakrishnan: ramki@mri.ernet.in E-mail addr ess B. Sah u: sahu@mri.ernet.in E-mail addr ess R. Thangadurai: thanga@mri.ernet.in

GUN B RAMAKRISHNAN B SAHU AND R THANGADURAI Abstra ct In this article shall study using elemen tary and com binatorial metho ds the distribution of quadratic nonresidues whic are not primitiv ro ots mo dulo or for an dd prime and is an in teger 1 In ID: 22808

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DISTRIBUTION OF QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI Abstra ct. In this article, shall study using elemen tary and com binatorial metho ds, the distribution of quadratic non-residues whic are not primitiv ro ots mo dulo or for an dd prime and is an in teger. 1. Intr oduction Distribution of quadratic residues, non-residues and primitiv ro ots mo dulo for an ositiv in teger is one of the classical problems in Num er Theory In this article, applying elemen tary and com binatorial metho ds, shall study the distribution of quadratic non-residues whic are not primitiv ro ots mo dulo dd prime ers. Let an ositiv in teger and an dd prime um er. denote the additiv cyclic group of order The ultiplicativ group mo dulo is denoted of order the Euler phi function. Denition 1.1. primitive ot mo dulo is generator of whenev er is cyclic. ell-kno wn result of C. F. Gauss sa ys that has primitiv ro ot if and only if or or for an ositiv in teger Moreo er, the um er of primitiv ro ots mo dulo these 's is equal to )) Denition 1.2. Let and in tegers suc that a; If the quadratic congruence (mo has an in teger solution x; then is called quadr atic esidue mo dulo Otherwise, is called quadr atic non-r esidue mo dulo n: Whenev er is cyclic and is primitiv ro ot mo dulo n; then for are all the quadratic non-residue mo dulo and for are all the quadratic residue mo dulo n: Also, for all suc that (2 )) are all the quadratic non-residues whic are not primitiv ro ots mo dulo n: or ositiv in teger n; set is primitiv ro ot mo dulo 2000 Mathematics Subje ct Classic ation. 11N69. Key wor ds and phr ases. quadratic non-residues, primitiv ro ots.

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI and is quadratic non-residue mo dulo Note that (1) (1) (2) and (2) When kno that and whenev er or or ha Also, it can easily seen that if then shall denote quadratic non-residue whic is not primitiv ro ot mo dulo QNRNP mo dulo n: Therefore, an is QNRNP mo dulo n: Recen tly zek and Somer [2 pro ed that i is either ermat prime (primes of the form 1) or or wice ermat prime. Moreo er, they pro ed that for if and only if or 18 or either or n= is equal to prime p; where 1) is also an dd prime. They also pro ed that when then In this article, shall pro the follo wing theorems. Theorem 1.1. et and any ositive inte gers. et or for any dd prime p: Then if and only if is either (i) or whenever +1 with is also prime or (ii) or whenever +1 is ermat prime. In this ase, the set is nothing but the set of al gener ators of the unique cyclic sub gr oup of or der +1 of When is not ermat prime, then it is clear from the ab discussion that := 1) When the natural question is that whether do es there exist an consecutiv pair of QNRNP mo dulo rom Theorem 1.1, kno that for all primes where is also prime um er. Theorem 1.2. et prime such that wher is also prime. Then ther do es not exist air of onse cutive QNRNP mo dulo p: In con trast to Theorem 1.2, shall pro the follo wing. Theorem 1.3. et any dd prime such that 1) Then ther exists air of onse cutive QNRNP mo dulo p: In the follo wing theorem, shall address eak er question than Theorem 1.3; but orks for arbitrary length Theorem 1.4. et any dd inte ger and inte gers. Then ther exists ositive inte ger dep ending only on and such that for every prime and (mo we have an arithmetic pr gr ession of length whose terms ar QNRNP mo dulo n; wher or Mor over, we an cho ose the ommon dier enc to QNRNP mo dulo n; whenever

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QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS 2. Preliminaries In this section, shall pro some preliminary lemmas whic will useful for pro ving our theorems. Prop osition 2.1. et any ositive inte ger and let or for any dd prime p: Then any inte ger is primitive ot mo dulo if and only if =q 6 (mo for every prime divisor of Pr of. Pro of is straigh forw ard and omit the pro of. The follo wing prop osition giv es criterion for QNRNP mo dulo whenev er or Prop osition 2.2. et any ositive inte ger. et any ositive inte ger of the form or wher is an dd prime. Then an inte ger is QNRNP mo dulo if and only if for some dd divisor of we have, (mo Pr of. Supp ose is QNRNP mo dulo n: Then, (mo If is ermat prime or wice ermat prime, then kno that ev ery non- residue is primitiv ro ot mo dulo n: Therefore, the assumption, is not suc um er. Th us there exists an dd in teger whic divides Since is not primitiv ro ot mo dulo n; Prop osition 2.1, there exists an dd prime dividing satisfying =q (mo Therefore, taking the square-ro ot of =q mo dulo n; see that (mo If (mo then taking the -th er oth the sides, it follo ws that is quadratic residue mo dulo p; con tradiction. Hence, get (mo or the con erse, let an in teger satisfying (mo (1) where is an dd divisor of Then squaring oth the sides of (1), conclude Prop osition 2.1 that cannot primitiv ro ot mo dulo n: By taking the -th er oth sides of (1), see that the righ hand side of the congruence is still as is dd and hence conclude that is quadratic non-residue mo dulo n: Th us the prop osition follo ws. Corollary 2.2.1. et prime. Supp ose is not ermat prime and divides If is QNRNP mo dulo p; then 1) is squar e-r ot of mo dulo for some dd divisor of

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI Pr of. By Lemma 2.2, it follo ws that there exists an dd divisor of suc that 1) (mo Since divides it is clear that 1) (mo and hence the result. Lemma 2.3. (K zek and Somer, [2 ]) et an dd ositive inte ger. Then (2 and (2 Theorem 2.4 (Brauer, [1]) et and ositive inte gers. Then ther exists ositive inte ger dep ending only on and such that for any artition of the set into -classes, we have ositive inte gers a; d; 1) and sd lie in only one of the 's. Using Theorem 2.4, Brauer [1] pro ed that for all large enough primes p; one can nd arbitrary long sequence of consecutiv quadratic residues (resp. non-residues) mo dulo p: Also, in series of pap ers, E. egh [3], [4 ], [5], [6 and [7] studied the distribution of primitiv ro ots mo dulo or 3. Pr oof of Theorem 1.1 Lemma 3.1. et and any ositive inte gers. et or for any dd prime p: Then if and only if is either (i) or whenever +1 with is also prime or (ii) or whenever +1 is ermat prime. Pr of. In the view of Lemma 2.3, it is enough to assume that Let where `; are ositiv in tegers suc that Case (i) 1) In this case, ha e, 1) and Hence, ould imply and Since the ositiv in teger satises ust prime um er. Therefore, those primes satises the yp othesis are of the form +1 where is also prime um er. Case (ii) 2) In this case, ha e, )) 1)) 1) 1) 1) No w, 1)

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QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS Therefore, implies and hence get, and Th us ha e, +1 whic ould imply cannot greater than If then ha +1 That is, if then the only in tegers satises the yp othesis are where is ermat prime. Con erse is trivial to establish. Pr of of The or em 1.1. Giv en that By Lemma 3.1, ha cases. Case (i) or where +1 where is also prime) Let an arbitrary elemen t. Then is quadratic non-residue mo dulo but not primitiv ro ot mo dulo n: Therefore Prop osition 2.2, kno that there exists an dd divisor of satises (mo Since +1 where is the only dd divisor of ust ha Therefore, (mo (mo +1 (mo Let the unique cyclic subgroup of Then with order of is +1 Hence as is arbitrary is the set of all generators of Case (ii) or where +1 is prime and is er of 2) Let Then Prop osition 2.2, kno that there exists an dd divisor of satisfying 1) (mo and hence +1 (mo Th us, where is the unique subgroup of of order +1 4. Pr oof of Theorem 1.2 Lemma 4.1. et prime such that wher is also prime. If a; 1) is air of QNRNP mo dulo p; then (mo Pr of. Giv en that and are QNRNP mo dulo p: Therefore, Prop osition 2.2, ha (mo and 1) 1) (mo That is, 1) (mo Hence the result. Pr of of The or em 1.2. By Lemma 3.1, kno that for these primes, there are exactly QNRNP mo dulo p: Supp ose assume that these QNRNP mo dulo are consecutiv pair, sa a; 1) Then Lemma 4.1, get, (mo end the pro of, shall, indeed, sho that is primitiv ro ot mo dulo

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI and arriv at con tradiction. pro is primitiv ro ot, ha to pro that the order of in is Since the order of is and the order of is equal to the order of it is enough to pro that is primitiv ro ot mo dulo p: By Prop osition 2.1, ha to pro that 6 (mo for ev ery prime divisor of In this case, ha and If then 1) =q and so 16 6 (mo as Hence, it is enough to pro that 6 (mo Indeed, the quadratic recipro cit la w, kno (mo and hence the theorem. 5. Pr oof of Theorem 1.3 Lemma 5.1. et prime such that et denote the total numb er of QNRNP mo dulo Then exactly 1) numb er of QNRNP mo dulo ar fol lowe by quadr atic non-r esidue mo dulo whenever wher is an dd inte ger; Otherwise, exactly half of QNRNP mo dulo is fol lowe by quadr atic non-r esidue mo dulo p: Pr of. First note that 1) 1) is dd if and only if 1) is dd if and only if where is an dd in teger. Let QNRNP mo dulo p: Let xed primitiv ro ot mo dulo p: Then there exists an dd in teger satisfying `; 1) and Therefore, is also QNRNP mo dulo p: Then ha e, (1 (mo This implies is quadratic residue mo dulo if and only if is quadratic non-residue mo dulo p: Therefore, to complete the pro of of this lemma, it is enough to sho that if is QNRNP mo dulo and 6 (mo then 6 (mo Supp ose not, that is, (mo Then, (mo Since it is clear that whic ould imply and therefore get (mo con tradiction and hence 6 (mo Similarly ha 6 (mo Note that (mo if and only if (mo 1) whic ould imply 1) as Since is dd, this happ ens precisely when where is an dd in teger. Hence the lemma. Pr of of The or em 1.3. Let an prime suc that 1) 1) If ossible, shall assume that there is no pair of consecutiv QNRNP mo dulo p: Let 1) Therefore, clearly By Lemma 5.1, kno that exactly half of QNRNP mo dulo follo ed quadratic non-residue mo dulo p: This implies, 1) um er of QNRNP mo dulo follo ed primitiv ro ots mo dulo p: Since there there are utmost 1) primitiv ro ots ailable, it follo ws that there exists QNRNP mo dulo follo ed QNRNP mo dulo p:

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QUADRA TIC NON-RESIDUES WHICH ARE NOT PRIMITIVE OOTS 6. Pr oof of Theorem 1.4 Giv en that is an dd in teger and is an in teger. Put and in Theorem 2.4. get natural um er dep ending only on and suc that for an -partitioning of the set ha ositiv in tegers a; d; d; 1) and whic are less than or equal to and lying in exactly one of the classes. Cho ose prime suc that (mo By Diric hlet's prime um er theorem on arithmetic progression, suc prime exists and there are innitely man suc primes. Let xed primitiv ro ot mo dulo Note that for eac there exists unique in teger 1) satisfying (mo partition the set in to parts as follo ws. if and only if (mo Since there exists an arithmetic progression of length sa a; d; 1) together with its common dierence lying in for some By the denition of our partition, ha id (mo and (mo where for eac satisfying (mo Since 's run through single residue class mo dulo can as ell assume, if necessary suitable translation, that (mo No w, ho ose an in teger suc that (mo 2) and (mo Then see that (mo Since is an dd in teger and are ev en in tegers, get, are dd in tegers together with (mo Therefore, divides the gcd( ; 1) Putting (mo get, a; d; 1) d; are QNRNP If is an dd in teger, then is also primitiv ro ot mo dulo If is an ev en in teger, then put whic is an dd in teger and hence it is primitiv ro ot mo dulo No the pro of is similar to case when and lea it to the readers. Before conclude this section, shall raise the follo wing questions. (1) Can Theorems 1.4 true for all large enough primes (2) What is the general prop ert of the set of all ositiv in tegers satisfying for an giv en ositiv in teger

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S. GUN, B. RAMAKRISHNAN, B. SAHU AND R. THANGADURAI Ac kno wledgmen t. are thankful to Professor M. zek for sending his pap er [2 ]. Also, are grateful to Professor D. Rohrlic oin ting out an error in the previous ersion of this pap er. References [1] A. Brauer, Ub er Sequenzen on otenzresten, Sitzungsb erichte der Pr eubischen kademie der Wissenschaften (1928), 9-16. [2] M. zek, L. Somer, necessary and sucien condition for the primalit of ermat um ers, Math. Bohem., 126 (2001), no. 3, 541-549. [3] E. egh, airs of consecutiv primitiv ro ots mo dulo prime, Pr c. mer. Math. So c. 19 (1968), 1169-1170. [4] E. egh, Primitiv ro ots mo dulo prime as consecutiv terms of an arithmetic progression, J. eine ngew. Math. 235 (1969), 185-188. [5] E. egh, Primitiv ro ots mo dulo prime as consecutiv terms of an arithmetic progression I, J. eine ngew. Math. 244 (1970), 108-111. [6] E. egh, note on the distribution of the primitiv ro ots of prime, J. Numb er The ory (1971), 13-18. [7] E. egh, Primitiv ro ots mo dulo prime as consecutiv terms of an arithmetic progression I, J. eine ngew. Math. 256 (1972), 130-137. School of Ma thema tics, Harish Chandra Resear ch Institute, Chha tna ad, Jhusi, Allahabad 211019, India. E-mail addr ess S. Gun: sanoli@mri.ernet.in E-mail addr ess B. Ramakrishnan: ramki@mri.ernet.in E-mail addr ess B. Sah u: sahu@mri.ernet.in E-mail addr ess R. Thangadurai: thanga@mri.ernet.in

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