ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS

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1 Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet of Mathematics Faculty of Arts ad Sciece Sakarya Uiversity Sakarya Turkey Abstract The study of umber sequeces has bee a source of attractio to the mathematicias sice aciet times. Sice the may of them are focusig their iterest o the study of the umbers. Udoubtely, triagular umbers are oe of these umbers. I this study, we deal with the relatios betwee square triagular umbers, special coditio of triagular umbers, ad balacig umbers. Also, we ivestigate positive iteger solutios of some Diophatie equatios such as ( x + y ) 8xy, ( x + y + ) 8xy, ( x + y) x( y ), ( x + y) x( y ), ( x + y ) 8xy +, x + y 6xy, x + y 6xy x 0, x 6xy + y x 0, ad other similar equatios related to square triagular ad balacig umbers. 00 Mathematics Subject Classificatio: D5, D, D7. Keywords ad phrases: triagular umbers, square triagular umbers, balacig umbers, cobalacig umbers, Diophatie equatios. Received November 9, Scietific Advaces Publishers

2 7 OLCAY KARAATLI ad REFİK KESKİN. Itroductio By a triagular umber, we mea the umber of the form T ( + ), where is a atural umber. A few of these umbers are, 3, 6, 0, 5,, 8, 36, 5, 55,..., ad so o. Also, it is well kow that x is a triagular umber, if ad oly if 8 x + is a perfect square. The -th triagular umber is formed usig a outher triagle, whose sides have dots. Similarly, square umbers ca be arraged i the shape of a square. The m-th square umber is formed usig a outher square, whose sides have m dots. The m-th square umber is S m m [ ]. Balacig umbers are umbers that are the solutios of the equatio ( m ) ( m + ) + ( m + ) + + ( m + r), (.) + callig r Z, the balacer correspodig to the balacig umber m. For example,, 6, 35, ad 0 are balacig umbers with balacers 0,,, ad 8, respectively. I what follows, we itroduce cobalacig umbers i a way similar to the balacig umbers. By modifyig (.), + we call m Z, a cobalacig umber if ( m ) + m ( m + ) + ( m + ) + + ( m + r), (.) + + for some r Z. Here, we call r Z, a cobalacer correspodig to the cobalacig umber m. A few of the cobalacig umbers are 0,,, ad 8 with cobalacers, 6, 35, ad 0, respectively [7], [8]. I this chapter, oe of the major questio we will be iterested i aswerig is whether or ot there is a close relatios betwee square triagular umbers ad balacig umbers. Sice triagular umbers are of the ( + ) form T ad square umbers are of the form S m, m square triagular umbers are iteger solutios of the equatio m ( + ) [ ]. We will get this equatio agai usig a amusig problem to see how square triagular umbers are obtaied from this problem. I Equatio (.), if we make substitutio m + r, we get

3 ON SOME DIOPHANTINE EQUATIONS RELATED ( m ) ( m + ) + ( m + ) + +. (.3) The this equatio gives us a problem as follows: I live o a street, whose houses are umbered i order,, 3,,, ; so the houses at the eds of the street are umbered ad. My ow house umber is m ad of course, 0 < m <. Oe day, I add up the house umbers of all the houses to the left of my house; the I do the same for all the houses to the right of my house. I fid that the sums are the same. So, how ca we fid m ad [3]? Sice m ( m + ) + + ( ) +, it follows that ( + ) Thus, we get m. Here, square umber. That is, ( m ) m ( + ) m( m + ). m is both a triagular umber ad a m is a square triagular umber. I Equatio (.3), sice m is a balacig umber, it is easy to see that a balacig umber is square root of a square triagular umber. For more iformatio about triagular, square triagular, ad balacig umbers, see [], [], [], ad [7]. Euler [] showed that a b is a triagular umber for a ( 3 + ) ad b ( 3 ). I 879 Roberts [], by usig Euler s formula, showed that the -th square triagular umber t is give by t ( + ) ( ). After the Subramaiam showed that u u u by takig t 6 u (see [5], [6]). Actually, the elemets of the sequece ( u ) are balacig umbers, which are obtaied from the solutios of the Equatio (.3). I this chapter, we will develop a method for fidig all square triagular umbers after usig well kow theorems. For this, we will use some Diophatie equatios, whose solutios are related to Pell, Pell-Lucas, ad the sequece ( v ), where v is give by

4 7 OLCAY KARAATLI ad REFİK KESKİN v Q. Before discussig about these sequeces, we itroduce two kids of sequeces amed geeralized Fiboacci ad Lucas sequeces { U } ad { V }, respectively. For more iformatio about geeralized Fiboacci ad Lucas sequeces, oe ca cosult [], [5], [6], [0], [], ad [8]. The geeralized Fiboacci sequece { U } with parameter k ad t, is defied by U 0 0, U, ad U + ku + tu for. Similarly, the geeralized Lucas sequece { V } with parameter k ad t, is defied by V 0, V k, ad V + kv + tv for, where k + t > 0. Also geeralized Fiboacci ad Lucas umbers for egative subscript are U V defied as U ( t) ad V ( t) for all N. Whe k ad t, we get U P ad V Q, where P ad Q are called Pell ad Pell-Lucas sequeces, respectively. Thus, P, P, ad + P + P P for ad Q 0, Q, ad Q Q Q + + for all N. Whe k 6 ad t, we represet U ad V by u ad v, respectively. Thus, u 0 0, u, ad u + 6u u ad v 0, v 6, ad v + 6v v for all. Now, we preset some well kow theorems regardig the sequeces ( P ) ( Q ), ad ( u ) without proof., Theorem.. Let γ ad δ be roots of the characteristic equatio x x 0. The, we have P γ δ ad Q γ + δ. Theorem.. Let α ad β be roots of the characteristic equatio x 6x + 0. The u α β ad v α + β. The formulas give i the above theorems are kow as Biet s formula. Let B deote -th balacig umber. From [8], we kow that

5 ON SOME DIOPHANTINE EQUATIONS RELATED 75 B ( ) ( 3 8 ) 8. From Theorems. ad., it is easily see that v Q. P u B ad The proof of the followig theorem is give i [3]. Theorem.3. Let Z[ ] { a + b a, b Z} ad γ +. The the set of uits of the rig [ ] Z is { γ Z}. Now, we metio some equatios, whose solutios are related to ( P ) ad ( Q ). Before discussig the equatios, we give some kow idetities for ( P ) ( Q ), ( B ), ad ( v ). ad, Well kow idetities for ( P ) ( Q ), ( B ), ad ( v ) are Q, 8 ( ), (.) P v 3B, (.5) P + + P Q, (.6) B 6B B + B, (.7) v v +. (.8) The followig theorem is a well kow theorem. We will give its proof for the sake of completeess. Theorem.. All positive iteger solutios of the Pell equatio Q x y are give by ( x, y), P with.

6 76 OLCAY KARAATLI ad REFİK KESKİN Q Proof. Assume that ( x, y), P. The ( x y )( x + y ) ad this shows that x y is a uit i Z [ ]. Moreover, sice x > 0 ad y > 0, we get x + y >. Therefore, there exists positive iteger + such that x y γ γp + P by Theorem.3. Sice γp + P ( + ) P + P P + P + P, we get ( x, y) ( P + P, P ). Thus, x P + P ( P + P ) ( P + P + P ) ( P + + P ) Q by idetity (.6). This shows that Q x ad y P. Q Coversely, if ( x, y) (, P ), the it follows from idetity (.) that x y. By usig (.) ad the above theorem, we ca give the followig corollaries: Corollary. All positive iteger solutios of the Pell equatio Q x y are give by ( x, y), P with. Corollary. All positive iteger solutios of the Pell equatio Q+ x y are give by ( x, y), P + with 0. We ow give the characterizatio of all square triagular umbers. Theorem.5. A atural umber x is a square triagular umber, if B ad oly if x, for some atural umber. Proof. Assume that x is square triagular umber. The ( + ) x m, for some atural umbers ad m. Next, multiplyig both sides by 8 ad rewritig the previous equatio, we obtai 8m + ( + ).

7 ON SOME DIOPHANTINE EQUATIONS RELATED 77 Let x + ad y m. The we get x y. Q v The by Corollary, it follows that ( x, y), P, B. v Therefore, + ad m B. This shows that x m B. Now B assume that x. By idetity (.5), we get + x B v (( v ) )(( v + ) ) (( v ) )(( v ) ). 3 Therefore, whe x B, x is a square triagular umber. This cocludes the proof.. Positive Solutios of Some Diophatie Equatios I this sectio, we cosider the equatios ( x + y ) 8xy, ( x + y + ) 8xy, ( x + y) x( y ), ( x + y ) 8xy +, x + y 6xy, x + y 6xy x 0, ad other similar equatios. The solutios of these equatios are related to square triagular umbers, balacig ad cobalacig umbers, ad the sequece ( y ), where y is give by y ( y + ) B. (( v ) ) (( v + ) ) By Theorem., it is kow that B. If we v y ( y + ) make substitutio y, the we get B. Here, a few terms of ( y ) sequece are, 8, 9, 88,..., ad so o. The followig lemma is give i [9]: Lemma. For, y 6y y. + +

8 78 OLCAY KARAATLI ad REFİK KESKİN Proof. Sice v y ad v 6v v with, we get 6y y + 6(( v ) ) (( v ) ) + (( 6v ) ) v (( v ) ) y. + + That is, y y y For,,, let b be -th cobalacig umber ad so let ( b ) deote the cobalacig umber sequece. The, the cobalacig umbers satisfy the similar recurrece relatio give i Lemma. The proof of the followig lemma is give i [7]. Lemma. For, b 6b b. + + Lemma 3. For every, y 8B ad y B B Proof. By idetities (.5) ad (.8), we get v v y B Therefore y B. Moreover, sice y y y, we get y ( y + y ) 6 ad therefore by usig y 8B, we fid + y + y + + y 6 8B + + 8B 6 8( B + + B ) Sice B B B B by idetity (.7), we get 8( B+ + B ) 8( 6BB+ + ) y + 8BB Therefore y B B. This cocludes the proof Now, we ca give the followig theorem. Sice its proof is easy, we ca omit it.

9 ON SOME DIOPHANTINE EQUATIONS RELATED 79 Theorem.. If is a odd atural umber, the Q is a eve atural umber, the y. Q y ad if At the itroductio of this study, we say that if x is a triagular umber, the 8 x + is a perfect square. Also, sice y B B Q+ ad y +, it follows that B B + is a triagular umber. Moreover by Lemma 3, it is see that y is odd iff is odd ad y is eve iff is eve. From ow o, we will try to fid all positive iteger solutios of some Diophatie equatios. Before givig these equatios, we preset the followig idetities that will be useful for fidig solutios. Sice the proof of the idetities is foud by usig Biet s formulas, Lemma, ad iductio method, we omit them. Theorem.. For every atural umber, we have the followig idetities: ad v + P + + Q +, (.) v P + Q+, (.) Q+ + P + B +, (.3) Q + P + B, (.) b y + B, (.5) b y+ B+. (.6) I [9], Potter showed that ( y y ) 8y y. By meas of this equatio, we fid other similar equatios ad we ivestigate solutios of them. Now we give the followig theorem:

10 80 OLCAY KARAATLI ad REFİK KESKİN Theorem.3. All positive iteger solutios of the Diophatie equatio ( x + y ) 8xy are give by ( x y) ( y, y ) with., + Proof. We kow that ( y + y+ ) 8y y+ from [9]. Therefore, ( x y) ( y, y ) is a solutio of the equatio ( x + y ) 8xy. Now, + assume that ( x + y ) 8xy for some positive itegers x ad y. A simple computatio shows that x y. The without loss of geerality, we may suppose that y > x. If we make substitutio u x + y ad v y x, the we get ( u ) ( u v ) ad therefore u u + u v. This shows that v ( u + ). The by Corollary, it u + Q+ follows that ( v, ) (, P + ) for some 0. Therefore, Q + u P + ad v. Sice u x + y ad v y x, we get u v u + v P + Q+ x ad y. This shows that x ad P + + Q+ y. By usig idetities (.) ad (.), we get v v x ad y +. This implies that x y ad y y. + By usig the above theorem, we ca give the followig three theorems i a similar way: Theorem.. All positive iteger solutios of the Diophatie equatio ( x + y ) 8xy are give by ( x y) ( y, y ) with., + Theorem.5. All positive iteger solutios of the Diophatie equatio ( x + y + ) 8xy are give by ( x, y) ( y +, y + ) with 0. +

11 ON SOME DIOPHANTINE EQUATIONS RELATED 8 Proof. Assume that ( x + y + ) 8xy for some positive itegers x ad y. Let we make substitutio u x ad v y. The we get ( u + v + 3) 8( u + )( v + ). Whe we rearrage the equatio, we get ( u + v ) 8uv. The the proof follows from Theorem.3. Coversely, if ( x, y) ( y +, y + ), + the a simple computatio shows that ( x + y + ) 8xy. Theorem.6. All positive iteger solutios of the Diophatie equatio ( x + y + ) 8xy are give by ( x, y) ( y +, y + ) with 0. + Now, we ca give the followig theorems, formed by meas of the above theorems, whose solutios are related to square triagular umbers, balacig ad cobalacig umbers, ad the sequece ( y ). Theorem.7. All positive iteger solutios of the equatio ( x + y) x ( y + ) are give by ( x y) ( B, B B ) or ( x, y) ( B, B B + ) with., + + Proof. Assume that ( x + y) x( y + ) for some positive itegers x ad y. The ( x + y) 6x( y + ) ad therefore ( x + y + ) 8.x( y + ). The by Theorem.3, it follows that ( x, y + ) ( y, y + ) or ( x, y + ) ( y +, y ) for some. Firstly, assume that ( x, y + ) ( y, y+ ) for some. Sice y is eve, the is also eve. Let k with k. Thus x y k ad y yk+. By usig Lemma 3, it follows that x B k ad y B k Bk+. Similarly, if ( x, y + ) ( y +, y ), we see that x Bk+ ad y B k Bk+. Coversely, if ( x y) ( B, B B ) or ( x y) ( B, B B ),, +, + + the with a simple computatio by usig idetity (.6), we get ( x + y) x ( y + ).

12 8 OLCAY KARAATLI ad REFİK KESKİN From the above theorem, we ca give the followig corollary: x + y Corollary 3. All positive iteger solutios of the equatio 6 xy x 0 are give by ( x y) ( B, B B ) or ( x, y) ( B, B B + ) with., + + Proof. x + y 6xy x 0 iff ( x + y) x( y + ) iff ( x + y).x(.y + ). The the proof follows from Theorem.7. Sice the proof of the followig theorem is similar to that of above theorem, we omit it. Theorem.8. All positive iteger solutios of the equatio ( x + y) x ( y ) are give by ( x, y) ( B B +, B ) or ( x, y) ( B B, B ) with k. k k+ + k+ + k k + + k From the above theorem, we ca give the followig corollary without proof: x Corollary. There is o positive iteger solutios of the equatio + y 6xy + x 0. Theorem.9. All positive iteger solutios of the equatio ( x + y + ) 8xy + are give by ( x, y) (,) or ( x, y) ( b +, b + ) with. Proof. Assume that ( x + y + ) 8xy + for some positive itegers x ad y. For x y, it is clear that ( x, y) (, ) is a solutio of the equatio ( x + y + ) 8xy +. The assume that x y. The without loss of geerality, we may suppose that x > y. Let u x + y ad v x y. The x u + v u v ad y. Sice ( x + y + ) 8xy +, u v we get ( u + ) 8( ) +. This shows that ( u ) v. By

13 ON SOME DIOPHANTINE EQUATIONS RELATED 83 Corollary, it is see that ( u, v) (, B ) for some. Thus v u + ad v B. Substitutig these values of u ad v ito the equalities x v u + v u v ad y, we get x u + v v + + B v + B + v + B + y + B +, ad y u v v + B v B + v B + y B +. That is, x y + B + ad y y B +. By idetities (.5) ad (.6), it follows that x b + ad y +. Coversely, if x b + ad y b +, the a simple computatio shows that ( b b + ) 8( b + )( b + ). This cocludes the proof. Sice the proof of the followig theorem is similar to that of above theorem, we omit it. Theorem.0. All positive iteger solutios of the equatio ( x + y ) 8xy + are give by ( x y) ( b, b ) with. b, Theorem.. All positive iteger solutios of the equatio x + y 6xy are give by ( x, y) ( B +, B ) with. Proof. Assume that x + y 6xy for some positive itegers x ad y. The ( x y) xy ad therefore x y. Without loss of geerality, we may suppose that x > y. If we make substitutio u x + y ad v x y, we get u + v u v x ad y. Sice x + y 6xy 0, it follows that ( x y) xy. Thus, if we

14 8 OLCAY KARAATLI ad REFİK KESKİN rewrite ew values of x ad y ito the equatio ( x y) xy, we get v ( u v ). This shows that u v. By Corollary, it Q+ follows that ( u, v) (, P + ) for some 0. Therefore u ad v P. Sice + Q + u + v u v x ad y, it is see that Q + + P + Q + + x ad P y. By usig idetities (.3) ad (.), we get x B + ad y B. Coversely, if ( x, y) ( B +, B ), the by idetity (.7), it follows that the proof. x + y 6. xy This cocludes x Theorem.. There is o positive iteger solutios of the equatio + y 6xy. Proof. Assume that x + y 6xy. The ( x y) xy ad therefore x y is a odd iteger. Let x + y u ad x y v. The, it ca be see that u v. Sice v is a odd iteger, the u v + 3 (mod 8), a cotradictio. This cocludes the proof. Theorem.3. All positive iteger solutios of the equatio x + y 6xy x 0 are give by ( x, y) ( B, BkBk + ) or ( x, y) k ( B B B ) with k. k+, k k+ Proof. Assume that x + y 6xy x 0 for some positive itegers x ad y. The ( x ) + ( y) 6( x)( y) ( x) 0. Let x a ad y b. The, it ca be see that a + b 6ab a 0. That is, ( a + b) a( b + ). Thus, by Theorem.7, there exists k such, k k k + that ( a b) ( B, B B ) or ( a b) ( B, B B ). Therefore,, k + k k + a B k, b B k Bk+ or a Bk +, b BkBk +. Sice a x ad

15 ON SOME DIOPHANTINE EQUATIONS RELATED 85 b y, it is see that, + x Bk y BkBk or x Bk +, y BkBk +., k k k +, k + k k + Coversely, if ( x y) ( B, B B ) or ( x y) ( B, B B ), it is easy to see that x + y 6xy x 0. x Theorem.. There is o positive iteger solutios of the equatio + y 6xy + x 0. Proof. Assume that x + y 6xy + x 0 for some positive itegers x ad y. The ( x ) + ( y) 6( x)( y) + ( x) 0. Thus, by Theorem.8, there exists k such that ( x, y) ( BkBk + +, Bk + ) or ( x, y) ( BkBk+ +, Bk+ + ), which is impossible. This cocludes the proof. Actually, we produce may equatios cosiderig from Potter s [9] showig ( y + y+ ) 8y y+. Now we give other similar equatios, whose solutios are related balacig ad cobalacig umbers more. While solvig these equatios, we agai use Pell equatios. Theorem.5. All positive iteger solutios of the equatio ( x + y ) 8xy + are give by ( x, y) ( B + b, B b ) with. Proof. Assume that ( x + y ) 8xy + for some positive itegers x ad y. The it follows that x y. Without loss of geerality, we may suppose x > y. Let u x + y ad v x y. Thus, we get ( u ) ( u v ) +. Whe we rearrage the equatio, it is see that u + v ( ). Thus by Corollary, there exists. such that u + Q Q ( v, ) (, P ). Therefore v ad u P. Sice Q v ad P B, we ca write v v ad u B. O

16 86 OLCAY KARAATLI ad REFİK KESKİN u + v the other had, sice u x + y ad v x y, it is see that x u v ad y. Now, if we write the values of u ad v ito the equalities u + v u v v x ad y, we get + 8B x ad 8B ( ) v y. By idetities (.5), (.6), ad the equality v y, it follows that x B + b ad y B b. Coversely, if ( x, y) ( B + b, B b ) with, the from idetities (.5), (.6), ad Lemma 3, it follows that ( x + y ) 8xy +. I a similar way, we ca give the followig theorem without proof: Theorem.6. All positive iteger solutios of the equatio ( x + y + ) 8xy + are give by ( x y) ( B + b +, B b ) with., Fially, we give two theorems, whose solutios are iterestig. Theorem.7. All positive iteger solutios of the equatio + y + x 0 are give by with. B + b + B b + (, ); is odd ( x, y) B b + + B b (, ); is eve x 6xy Proof. Assume that x 6xy + y + x 0. The, x 6xy + y + x 0 iff ( x + y) x( y ) + for some atural umbers x ad y. Next, whe we multiply both sides of the equatio ( x + y) x ( y ) + by ad rewrite previous equatio, we get

17 Let ON SOME DIOPHANTINE EQUATIONS RELATED 87 ( x + y + ) 8.x( y ) +. u x ad v y. The we get ( u + v + ) 8uv +. Sice all positive iteger solutios of the equatio ( u + v + ) 8uv + are ( u v) ( B + b +, B b ) or ( u, v) ( B b, B + b + ) by, Theorem.6, it is see that u B + b + ad B b or B b v u ad v B + b +. Thus, we get x ( B + b + ) ad y ( B b + ) or x ( B b ) ad y ( B + b + ). It is well kow that B is eve iff is eve ad b is always eve. The we get B + b + B b + (, ); is odd ( x, y). B + + b B b (, ); is eve B Coversely, if (, ) ( + b + B, b x y + ), where is odd B b B + b + ad ( x, y) (, ), where is eve; by usig idetities (.5), (.6), ad Lemma 3, it ca be show that x 6xy + y + x 0. This cocludes the proof. Now we ca give the followig theorem easily: Theorem.8. All positive iteger solutios of the equatio + y x 0 are give by x 6xy ( x, y) B b B + b (, ); is odd B + b B b (, ); is eve with.

18 88 OLCAY KARAATLI ad REFİK KESKİN Refereces [] A. Behera ad G. K. Pada, O the square roots of triagular umbers, Fiboacci Quart. 37() (999), [] L. E. Dickso, History of the Theory of Numbers, Volume, Strechert, New York, 97. [3] G. H. Hardy ad E. M. Wright, A Itroductio to the Theory of Numbers, Fifth Editio, Oxford Uiversity Press, New York, 979. [] D. Kalma ad R. Mea, The Fiboacci umbers-exposed, Mathematics Magazie 76 (003), [5] R. Keski ad B. Demirtürk, Solutio of some Diophatie equatios usig geeralized Fiboacci ad Lucas sequeces, Ars Combiatorica (accepted). [6] J. B. Muskat, Geeralized Fiboacci ad Lucas sequeces ad rootfidig methods, Mathematics of Computatio 6(03) (993), [7] G. K. Pada ad P. K. Ray, Cobalacig umbers ad cobalacers, Iteratioal Joural of Mathematics ad Mathematical Scieces 8 (005), [8] G. K. Pada, Some fasciatig properties of balacig umbers, Fiboacci Numbers ad their Applicatios 0 (006). [9] D. C. D. Potter, Triagular square umbers, The Mathematical Gazette 56(396) (97), [0] S. Rabiowitz, Algorithmic Maipulatio of Fiboacci Idetities, Applicatio of Fiboacci Numbers, Volume 6, Kluwer Academic Publ., Dordrect, The Netherlads, (996), [] P. Ribeboim, My Numbers, My Frieds, Spriger-Verlag, New York, Ic., 000. [] M. R. Schroeder, Number Theory i Sciece ad Commuicatio, Spriger, 005. [3] S. A. Shirali, Fu with Triagular Numbers, 00, [] Staford Math Circle, Suday, May 9, 00, Square-Triagular Numbers, Pell s Equatio ad Cotiued Fractios, [5] K. B. Subramaiam, A simple computatio of square triagular umbers, Iteratioal Joural of Mathematics Educatio i Sciece ad Techology 3 (99),

19 ON SOME DIOPHANTINE EQUATIONS RELATED 89 [6] K. B. Subramaiam, A divisibility properties of square triagular umbers, Iteratioal Joural of Mathematics Educatio i Sciece ad Techology 6 (995), [7] James J. Tattersall, Elemetary Number Theory i Nie Chapters, Cambridge Uiversity Press, 0. [8] Z. Yosma ad R. Keski, Some ew idetities cocerig geeralized Fiboacci ad Lucas umbers (submitted). g