#A62 INTEGERS 16 (2016) REPRESENTATION OF INTEGERS BY TERNARY QUADRATIC FORMS: A GEOMETRIC APPROACH

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1 #A6 INTEGERS 16 (016) REPRESENTATION OF INTEGERS BY TERNARY QUADRATIC FORMS: A GEOMETRIC APPROACH Gabriel Durha Deartent of Matheatics, University of Georgia, Athens, Georgia gjdurha@ugaedu Received: 9/11/15, Revised: //16, Acceted: 8/1/16, Published: 8/6/16 Abstract In 1957 NC Ankeny rovided a new roof of the three suares theore using geoetry of nubers This aer generalizes Ankeny s techniue, roving exactly which integers are reresented by x + y + z and x + y + z as well as roving su cient conditions for an integer to be reresented by x + y + z and x + y + 7z 1 Introduction A natural uestion in the study of uadratic fors concerns the reresentation of integers by certain uadratic fors Given an n-ary uadratic for Q and an integer, does there exist a vector ~x Z n such that Q(~x)? This aer considers this uestion for four ternary uadratic fors: x +y +z, x +y +z, x +y +z, and x +y +7z We are able to rove exactly which integers are reresented by the first two fors and rovide su cient conditions for an integer to be reresented by the final two fors In 191, Burton Jones [] rovided one of the key breakthroughs in the study of ositive definite ternary uadratic fors when he roved that the fors of a given genus collectively reresent all ositive integers not ruled out by certain congruence conditions As a result, if a uadratic for is alone in its genus then one can show it reresents an integer by showing that it locally reresents While Jones work ade the task of deterining which integers are reresented by our first three fors 1 ore straightforward, the roofs resented in this aer ake no use of Jones result Furtherore, Jones result tells us little about fors with one or ore genus-ates While ternary uadratic fors have been studied for centuries, relatively little is known about the reresentation of integers by fors not alone in their genus In articular, the uestion of exactly which integers are 1 As well as all other class nuber one fors (ie fors which are alone in their genus)

2 INTEGERS: 16 (016) reresented by x + y + 7z reains oen, desite the fact that this for has the sallest deterinant (and thus, in a sense, is the ost sile) of any classically integral ternary uadratic for not alone in its genus While this aer does not fully answer the uestion, we do rovide a new roof of a result of Kalansky [4] in our roof of Theore (a) For a ore corehensive list of what is known about the reresentation of integers by x + y + 7z the reader ay consult Wang and Pei [5] To rove our results we generalize a ethod develoed by NC Ankeny [1], which he used to rovide a new roof of the Gauss-Legendre three suares theore Ankeny began with a ositive integer not of the for 4 k (8` + 7) for soe k, ` Z and, using Dirichlet s theore on ries in an arithetic rogression, defined a rie based on the rie factors of He then defined a linear a, ~ : (x, y, z) 7! (R, S, T ) By considering the body {(R, S, T ) R R + S + T < }, he was able to invoke Minkowski s convex body theore to guarantee integer values of x, y, z such that ~ (x, y, z) By the roerties of the transforation ~, he shows that R + S + T and R Z To colete the roof of the three suares theore Ankeny showed that S + T is a su of two integer suares by showing that for all ries dividing S + T to an odd ower, 1 1 We alter this ethod by changing the transforation ~ to show that S + T is reresented by other binary fors We obtain the following results: Theore 1 The uadratic for x + y + z reresents a ositive integer if and only if is not of the for 4 k (8` + 7) for soe k, ` Z Theore The uadratic for x + y + z reresents a ositive integer if and only if is not of the for 4 k (16` + 14) for soe k, ` Z Theore (a) If is a ositive integer of the for 4 k (8` + 5) for soe k, ` Z and ord 7 () is even then is reresented by the uadratic for x + y + 7z (b) If is a ositive integer of the for 4 k (8` + 1) for soe k, ` Z and ord () is even then is reresented by the uadratic for x + y + z Background The following section serves as a brief introduction to the aterial resented in the aer Let n N An n ary integral uadratic for, Q, is a hoogeneous olynoial

3 INTEGERS: 16 (016) of degree two, ie, Q : Z n! Z Q : (~x) 7! X 1aleialejalen a i,j x i x j, where a i,j Z for all 1 ale i ale j ale n We say Q reresents an integer if there exists an ~x Z n such that Q(~x) A uadratic for Q is ositive definite if both of the following hold: (i) Q(~x) 0 for all ~x Z n, (ii) Q(~x) 0 if and only if ~x ~0 Henceforth, by for we ean ositive definite ternary integral uadratic for a Throughout the aer b refers to the Jacobi sybol Furtherore, a uadratic for Q is ultilicative if the following holds: for all integers a and b, if Q reresents a and b, then Q reresents ab The following roofs rely on three external theores, two of which, Dirichlet s theore on ries in an arithetic rogression and Minkowski s theore on convex syetric bodies, are soe of the ore owerful and well-known results of nineteenth century nuber theory The third theore coes fro the theory of binary uadratic fors and was reroved using the geoetry of nubers by Clark, Hicks, Parshall, and Thoson in 01 and states the following: Theore 4 ([]): For c, and 7, x + cy reresents a ositive integer if and only if 1 for all ries dividing to an odd ower c Proofs of Theores For Theores 1 and we can look at these fors (od 8) and (od 16) and see that they do not reresent any ositive integer congruent to 7 (od 8) and 14 (od 16), resectively Thus, the roofs of these theores will rovide necessary and su cient conditions for a ositive integer to be reresented by these fors We can also see that for any ositive Z, if x + y + z reresents, then x + y + z reresents, and if x + y + z reresents, then x + y + z reresents Thus, Theores 1 and can be roven sily by showing which ositive odd integers are reresented by these fors; however, the urose of this aer is to dislay the anner in which Ankeny s techniue can be generalized In the sirit of this urose we coletely rove Theore 1 using Ankeny s techniue We also use Ankeny s techniue to show which ositive odd integers are reresented by the for x +y +z, but, for brevity s sake, we refer to Theore 1 when roving

4 INTEGERS: 16 (016) 4 which even integers are reresented by the for and leave the Ankeny-style roof to the reader 1 x + y + z Here we wish to rove that the uadratic for x + y + z reresents all ositive integers not of the for 4 k (8` + 7) for soe k, ` Z Proof of Theore 1 Let be a ositive integer which cannot be written as 4 k (8` + 7) for any k, ` Z Without loss of generality we can assue that is suarefree We will first consider the case where (od 8) Let be an odd rie such that >, 1 (od 8), and 1 for all ries (we know such a exists by Dirichlet s theore on ries in an arithetic rogression) This construction of guarantees that there exists a t Z with t 1 1 Y (od ) This construction of also guarantees Y Y Y Thus there exists a b Z, where b is odd, such that b (od ) As a result, there is an h 1 Z such that b h 1 Looking at this stateent odulo we get h 1 0 (od ) Thus h 1 h for soe h Z Therefore we have that b h We now consider the body,, defined by {(R, S, T ) R R + S + T < }, where R tx + bty + z S x + b y T y Note that vol( ) ( 4 )( )( 1 )( ) 8 / Let ~ : R! R where ~ : (x, y, z) 7! (R, S, T ) be the function associated to the above transforation We see that ~ (~x) M ~ ~x where M ~ 6 4 t bt b

5 INTEGERS: 16 (016) 5 By looking at ~ as a linear a we see that M ~ is the standard atrix of ~ Furtherore, det(m ~ ) ()( )( ) / We now wish to exaine ~ 1 ( ) Since is a convex syetric body and ~ is invertible, we know that ~ 1 ( ) is convex and syetric as well We see that vol( ~ 1 ( )) 1 det(m ~ ) vol( ) 1 vol( ) 8 / > By Minkowski s theore on convex syetric bodies there exist x 1, y 1, z 1 Z, not all zero, such that ~ (x 1, y 1, z 1 ) Let (R 1, S 1, T 1 ) : ~ (x 1, y 1, z 1 ) We see that and also that R 1 + S 1 + T 1 (tx 1 + bty 1 ) + ( x 1 + by 1 ) + ( t (x 1 + by 1 ) + 1 ((x 1 + by 1 ) + y 1) (x 1 + by 1 ) + 1 (x 1 + by 1 ) 0 (od ), S 1 + T 1 1 ( x 1 + bx 1 y 1 + (b + )y 1) 1 ( x 1 + bx 1 y 1 + hy 1) x 1 + bx 1 y 1 + hy 1 y1 ) While S 1, T 1 6 Z, we can see that S1 + T1 Z Thus R1 + S1 + T1 Z, 0 < R1 + S1 + T1 <, and R1 + S1 + T1 0 (od ) We can now conclude that R1 + S1 + T1 Our goal now is to show that R1 + S1 + T1 is reresented by the uadratic for x + y + z Since R 1 Z we wish to show that S1 + T1 is of the for a + b for soe a, b Z By the theore of Clark, Hicks, Parshall, and Thoson, it will su ce to show that 1 for all ries dividing S1 + T1 to an odd ower [] We note here that x + y is ultilicative and reresents, so we need not consider ; furtherore, S1 + T1 ale <, and so we need not consider Thus, it will su ce to show 1 for all odd ries v, where v : (S1 +T1 ) We will do this in two cases Case 1 - : We see that 0 6 R1 (od ); it follows that R1 (od ) and so Furtherore, 0 v (x 1 + by 1 ) + y1 (od ) so (x 1 + by 1 ) ( )y1 (od ) Since divides v to an odd ower we know y (od ) and thus we can conclude 1

6 INTEGERS: 16 (016) 6 Case : Since is assued suarefree, we can assue - Since R 1 + S 1 + T 1, we know R 1 0 (od ) and R 1 Furtherore, (x 1 + by 1 ) + y (x 1 + by 1 ) 0 (od ), so (x 1 + by 1 ) Hence, R 1 1 ( (x1+by1) + y 1) It follows that y 1 (od ) Thus, 1 In both cases + S 1 +T 1 R 1 + 1, thus there exist a, b Z such that S1 + T1 a + b Therefore, is reresented by the for x + y + z, as reuired For odd: If 1, 5 (od 8) take to be an odd rie such that >, 1 (od 8), and 1 for all ries We see that 1 Y 1 Y Y Y (od ) and take the follow- We now roceed as before; however, we take t 1 4 ing transforation R tx + bty + z S x + b y T y For even: Take 1 Since is assued suarefree we can assue that 1 is odd If 1 1, (od 8) take to be an odd rie such that 1 (od 8), > 1, and 1 Y 1 1 for all 1 We see that Y 1 Y 1 Y Now take t, b Z such that t 1 (od 1) and b 1 (od ), where b is even Thus there exists an h Z such that We now consider the body defined by b h 1 {(R, S, T ) R R + S + T < 1 }

7 INTEGERS: 16 (016) 7 (note that vol( ) 16 / 1 ) where R tx + bty + 1 z S x + b y T 1 y We define ~ to be the function associated to the above transforation We see that ~ is an invertible linear a and the deterinant of its standard atrix is / 1, thus the volue of ~ 1 ( ) is 16 > 16 () We now consider the lattice {(x, y, z) Z x 0 (od )} Since [Z : ], we know the fundaental doain of has volue and thus we can invoke Minkowski s theore on convex syetric bodies to show there exists a nonzero oint (x 1, y 1, z 1 ) such that ~ (x 1, y 1, z 1 ) Let (R 1, S 1, T 1 ) : ~ (x 1, y 1, z 1 ) We see that R 1 + S 1 + T 1 t (x 1 + by 1 ) + 1 (x 1 + by 1 ) + 1 ( 1 y 1 ) t (x 1 + by 1 ) + 1 ((x 1 + by 1 ) + y 1) 1 (x 1 + by 1 ) + 1 (x 1 + by 1 ) 0 (od 1 ) and that 1 (S 1 + T 1 ) 1 ((x 1 + by 1 ) + 1 y 1) 1 ( x 1 + bx 1 y 1 + hy 1) x 1 + bx 1y 1 + hy 1 Since x 1 is even, we see that S 1 +T 1 Z Moreover, 0 < R1 + S 1 +T 1 < 1, and we conclude that R1 + S 1 +T 1 1 We now define v : (S1 +T1 ) We now wish to show that S 1 +T 1 is reresented by the binary uadratic for x + y As before, it will su ce to show 1 for all odd ries dividing v to an odd ower [] We roceed in two cases Case 1 - : Since 1 R1 (od ), we have 1 1 Furtherore, (x 1 + by 1 ) 1 y1 (od ) It follows that 1 1

8 INTEGERS: 16 (016) 8 Case : By the sae arguent resented in the (od 8) case, we see that y1 (od ) By the construction of, 1 In both cases we have 1 Therefore there exist a 1, b 1 Z such that S 1 +T 1 a 1 + b 1 Since the for x + y is ultilicative and reresents we also know there exist a, b Z such that S1 + T1 a + b Furtherore, since R1+ S 1 +T 1 1, R1+S 1+T 1 Noting that R 1 Z we see R1+a +b Therefore is reresented by x + y + z, as reuired If 1 5 (od 8), take to be an odd rie such that > 1, 5 (od 8), and 1 for all ries 1 We see that 1 Y 1 ( 1) Y 1 Y 1 ( 1) 1 ( 1) Y 1 1 The rest of the roof roceeds as above If 1 7 (od 8), take to be an odd rie such that > 1, (od 8), and 1 for all ries 1 We see that 1 Y 1 Y 1 1 The rest of the roof roceeds as above This coletes the roof of Theore 1 x + y + z ( 1) Y 1 1 Y 1 We now wish to rove that the uadratic for x + y + z reresents all ositive integers not of the for 4 k (16` + 14) for soe k, ` Z Proof of Theore Let be a ositive integer which cannot be written as 4 k (16` + 14) for any k, ` Z We wish to rove that is reresented by the for x + y + z Without loss of generality we can assue that is suarefree As before we first consider the case where (od 8) Let be an odd rie where >, 1 (od 8), and 1 for all ries Furtherore, this construction of ensures that there exists a t Z such that t 1 (od ) We also have

9 INTEGERS: 16 (016) 9 1 Y Y Y Y Thus there exists a b Z, where b is even, such that b (od ) As a result we know b h 1 for soe h 1 Z We can thus write h 1 h for soe h Z and see b h We now consider the body defined by {(R, S, T ) R R + S + T < } (note that vol( ) 4 8 / ) where R tx + bty + z S x + b y T y Let ~ : R! R, where ~ : (x, y, z) 7! (R, S, T ), be the function associated to the above transforation Thus, ~ (~x) M ~ ~x where M ~ t bt 6 4 b Since is a convex syetric body, ~ ( ) is as well We see that vol( ~ 1 ( )) vol( ) det(m ~ ) 8 > 8 Minkowski s theore on convex syetric bodies guarantees the existence of x 1, y 1, z 1 Z (not all zero) such that ~ (x 1, y 1, z 1 ) Let (R 1, S 1, T 1 ) : ~ (x1, y 1, z 1 ) We see that R1 + S1 + T1 (tx 1 + bty 1 ) + ( x 1 + by 1 ) y1 + ( ) t (x 1 + by 1 ) + 1 ((x 1 + by 1 ) + y 1) t (x 1 + by 1 ) + 1 (x 1 + by 1 ) 0 (od ),

10 INTEGERS: 16 (016) 10 and that S 1 + T 1 1 (4 x 1 + 4bx 1 y 1 + (b + )y 1) 1 (4 x 1 + 4bx 1 y 1 + hy 1) x 1 + bx 1 y 1 + hy 1 We now see that S1 +T1 Z and, as before, we can conclude that R1+S 1 +T1 Let v : (S1 + T1 ) We wish to show that S1 + T1 is of the for a + b for soe a, b Z As was the case in our roof of Theore 1, it will su ce to show 1 for all odd ries dividing v to an odd ower (and we need not consider ) [] We will do so in two cases Case 1 - : Since R 1 (od ), 1 Furtherore, (x 1 + by 1 ) y1 (od ) Thus 1 Case : Since is assued suarefree we know that - Since R 1 + S 1 + T 1, R 1 0 (od ) and so R 1 Moreover, we know that (x 1 + by 1 ) y 1 (x 1 + by 1 ) 0 (od ) Thus (x 1 + by 1 ) Noting that R ( (x1+by1) + y 1), we see that 1 ( y 1) (od ), and so y1 (od ) Thus 1 In both cases we have 1 for all odd ries dividing v to an odd ower We now know there exist a, b Z such that S1 + T1 a + b Therefore, is reresented by the uadratic for x + y + z, as reuired For odd we consider two situations For 7 (od 8), take to be an odd rie such that >, (od 8), and 1 Y 1 for all ries We see that Y ( 1) Y Y The rest of the roof roceeds as above For 1, 5 (od 8), take to be an odd rie such that >, 1 (od 8), and 1 for all ries We see that 1 Y 1 Y Y Y

11 INTEGERS: 16 (016) 11 We let t 1 (od ) and take the following transforation R tx + bty + z S x + b y T y The rest of the roof roceeds as above For even we let 1 By Theore 1 we know, for 1 1,, 5 (od 8), there exist x, y, z Z such that 1 x +y +z Thus 1 x +(y) +(z) Therefore, is reresented by x + y + z This coletes the roof of Theore x + y + 7z and x + y + z Here we wish to show that if is a ositive integer of the for 4 k (8` + 5) for soe k, ` Z and ord 7 () is even then is reresented by the uadratic for x + y + 7z and if is a ositive integer of the for 4 k (8` + 1) for soe k, ` Z and ord () is even then is reresented by the uadratic for x + y + z Proof of Theore We first seek to rove Theore (a) Let be a ositive integer of the for 4 k (8` + 5) for soe k, ` Z with ord 7 () even Without loss of generality we can assue is suarefree (and conseuently that 7 - ), and 5 (od 8) Let be an odd rie such that >, 1 (od 8), and 1 for all ries By this construction of, there exists a t Z such that t 1 4 (od ) Additionally we see that 1 Y 1 Y Y Y 7 Thus there is a b Z, where b is odd, such that b 7 (od ) This shows us that b h 1 7 for soe h 1 Z Looking at this stateent odulo 4 we see that h 1 0 (od 4) Therefore h 1 4h for soe h Z and so b 4h 7 Looking at the above stateent odulo 8 we see 4h 4 (od 8), thus h is odd We now consider the body defined by {(R, S, T ) R R + S + T < }

12 INTEGERS: 16 (016) 1 (note that vol( ) 4 8 / ) where R tx + bty + z S x + b y T 7 y Let ~ : R! R, where ~ : (x, y, z) 7! (R, S, T ), be the function associated to the above transforation Thus, ~ (~x) M ~ ~x where M ~ t bt 6 4 b We see that det(m ~ ) 7 / Since is a convex syetric body, ~ 1 ( ) is as well It follows that vol( ~ 1 ( )) vol( ) det(m ~ ) 16 7 > 8 Minkowski s theore on convex syetric bodies guarantees that there exist x 1, y 1, z 1 Z, not all zero, with ~ (x 1, y 1, z 1 ) Let (R 1, S 1, T 1 ) : ~ (x 1, y 1, z 1 ) We see that R 1 + S 1 + T 1 (tx 1 + bty 1 ) + ( x 1 + by 1 ) + ( 7y1 ) t (x 1 + by 1 ) ((x 1 + by 1 ) + 7y 1) t (x 1 + by 1 ) (x 1 + by 1 ) 0 (od ), and that S 1 + T (4 x 1 + 4bx 1 y 1 + (b + 7)y 1) 1 4 (4 x 1 + 4bx 1 y 1 + 4hy 1) x 1 + bx 1 y 1 + hy 1 Thus S1 + T1 Z and, as before, we conclude R1 + S1 + T1 Since R 1 Z we wish to show S 1 + T1 is of the for a + 7b for soe a, b Z It will be su cient to show that 1 for all ries dividing S1 + T1 to an odd ower [] 7

13 INTEGERS: 16 (016) 1 Note that ale <, so we can assue 6 Furtherore, if S1 + T1 x 1 + bx 1 y 1 + hy1 then x, y (since h and b are odd), thus divides S1 + T1 to an even ower and so we need not consider We also need not consider 7, as the for x + 7y reresents 7 and is ultilicative As a result it will su ce 7 to show 1 for all ries v, where 6, 7, and v : 4(S1 + T1 ) We roceed in two cases Case 1 - : We see that (x 1 + by 1 ) 7y1 (od ) and R 1 (od ) It follows that 1 7 Case : Note that since is assued suarefree, - Thus, R 1 0 (od ) and so R 1 Additionally, (x 1 + by 1 ) + 7y 1 (x 1 + by 1 ) 0 (od ) so (x 1 + by 1 ) We now see that R ( (x1+by1) +7 y 1) Thus 1 4 (7 y 1) (od ) It follows that 7y 1 4 (od ) By the construction 7 of, 1 7 In both cases we see 1 Thus there exist a, b Z such that a +7b Therefore, is reresented by x + y + 7z, as reuired This coletes the roof of Theore (a) The roof of Theore (b) follows the above roof; however, we take to be an odd rie such that >, 1 (od 1), and 1 for all ries It follows that 1 Y 1 Y Y Y and we take the following transforation R tx + bty + z S x + b y T y This coletes the roof of Theore Acknowledgeent: This research was suorted by the National Science Foundation (DMS ) I would additionally like to thank Dr Katherine Thoson, Dr Jerey Rouse, Dr Pete Clark, Sarah Blackwell and Ti any Treece for their suort I would finally like to thank Dr Theodore Shifrin for his guidance over the last three years and his decades of service to the University of Georgia and its students

14 INTEGERS: 16 (016) 14 References [1] NC Ankeny Sus of three suares, Proceedings of the Aerican Matheatical Society 8 (1957), [] PL Clark, J Hicks, H Parshall, and K Thoson GoNI: Pries reresented by binary uadratic fors, Integers 1 (01), A7, 18 [] B Jones The regularity of a genus of ositive ternary uadratic fors, Trans Aer Math Soc (191), [4] I Kalansky The first nontrivial genus of ositive definite ternary fors, Matheatics of Coutation 64 (1995), [5] X Wang and D Pei Eisenstein series of / weight and one conjecture of Kalansky, Sci China Ser A 44 (001) no 10,