The four square theorem was proved by Lagrange in 1770; every positive integer is the sum of four squares. i.e, n=A2+B2+C2+D2, where, A,B,C,D ϵ z An interesting proof is presented here based on Hurwitz integers, a subset of Quarternions which act like integers in four dimension and have the prime divisor propert

R, RAMYA

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A STUDY ON FOUR SQUARE THEOREM (R.RAMYA,Dr.S.SANGEETHA,M.MAHALAKSHMI,P.PRIYA) (ramya25071987@gmail.com,sangeethasankar2016@gmailcom ) Department of Mathematics Dhanalakshmi Srinivasan College of Arts and Science for Women (Autonomous) Perambalur      ABSTRACT:  The  four  square  theorem  was  proved by   Lagrange  in  1770; every  positive  integer  is  the  sum  of  four squares. i.e, n=A2+B2+C2+D2,      where, A,B,C,D ϵ z An interesting proof is presented here based on Hurwitz integers, a subset of Quarternions which act like integers in four dimension and have the prime divisor property.  KEYWORDS: Geometric Number, probabilities number, analytical number theory. INTRODUCTION: Number theory, known as the queen of mathematics is the branch of mathematics that concerns about the positive integers 1, 2, 3, 4, 5 which are often called natural numbers and their appealing properties from antiquity, these natural numbers classified as odd numbers, even numbers, square numbers, prime numbers, Fibonacci numbers, triangular numbers, etc. Due to the dense of unsolved problems, number theory plays a significant role in mathematics. The  four  square  theorem  was  proved by   Lagrange  in  1770; every  positive  integer  is  the  sum  of  four squares. i.e, n=A2+B2+C2+D2,       where, A,B,C,D ϵ z An interesting proof is presented here based on Hurwitz integers, a subset of Quarternions which act like integers in four dimension and have the prime divisor property.  DEFINITION: QUARDRATIC FORM: A binary quardratic form  2 +     +   2,   Where A,B,C∈ ℤ, can be viewed as an integer –valued function of integer  pairs, or vectors ( ,  ).                    Example,  2 2 + 3   + 4 2 Where,  A= 2 ,B= 3, C=4 DEFINITION:  HURWITZ INTEGER:  The  quaternions  in ℤ [1 + + +  ,i ,j ,k ]  are  called  the  Hurwitz   Integers. DEFINITION:  DIOPHANTINE EQUATION:  A equation of the form a2 + b2 = c2where a, b and c are integers is known as a diophantine equation.     FOUR  SQUARE  THEOREM:  DEFINITION:  Every  natural  number  can  be  represented  as  the  sum  of  four  integer  square.     P = a02 +  a12 + a22 + a32   Where  the  four  numbers  a0 , a1 , a2, a3  are  integers.  For   1 , 2 ,3 , 4  can be  represented  as  the  sum  of  four  square  theorem.    EXAMPLES  1 = 12 + 02 + 02 + 02 2 = 12 + 12 + 02 + 02 3 =  12 + 12 + 12 + 02 4 = 12 + 12 + 12 + 12   (or)   22 +02 +02 +02 REAL MATRICES AND C : In this chapter we introduce  4 –dimensional  “Hypercomplex numbers” called quaternion. A  quaternion  is easily defined as a 2×2 matrix of complex numbers ,  but to see why we might expect matrices to behave like numbers, we first show to model the complex numbers by 2×2 real matrices. For each a +bi∈∁, with real a and b, consider the matrix      M(a +bi) = (    )  −    It is easy to check that     M(a1 + b1i) + M(a2 + b2i) = M( ( a1 + a 2+(b1 + b2)i))                                                = M( (a1 + b1i) + (a2 + b2i))  M(a1 + b1i)  M(a2 + b2i)   =M (a1a2 – b1b2 +(a1b2 + b1a2)i)                       = M( (a1 + b1i)( a2 + b2i)).    Thus matrix sum and product correspond to complex sum and product , and therefore the matrices   (   )     for  a,b ∈ ℝ  −    .Behave exactly like  the  complex  numbers   a + bi.        Another way to see this is to write   (    )     = a (1  −    0 The identity matrix  ,                                             1 = (1 0)  0 1 Behaves like the number 1 , and      i =  ( 0 1)  −1 0 0 + b ( 0 ) 1 −1 1 = a1 + bi.  ) 0  Behaves like . Indeed    i2 = (−1 0 0  = -1. ) −1 Not only does this matrix representation of  ∁ have natural counterparts of 1 and i , it also has a natural interpretation of the norm on ∁ as the determinant . this is so because            norm(a + bi) = a2 + b2 = det  (    )  −    The multiplicative property of the norm follows from the multiplicative property of the determinant;   det   (  1  1) det (  2  2) =    − 1  1 − 2  2                            det  (  1  1) (  2  2)            →  (*)   − 1  1 − 2  2 And  since  the  matrix  product  on  the  right-hand  side  equals  ( 1 2 − 1 2  1 2 +  1 2)  − 1 2 − 1 2  1 2 − 1 2 Equation (*) gives a new way to derive the diophantus two square identity .           Replacing each det (   )in (*) by a2 + b2  we get   −       (a12 + b12)(a22 +b22) = (a1a1 – b1b2)2 + (a1b2 + b1a2)2.   THE  FOUR  SQUARE  IDENTITY:            If q =  a1 + bi + cj + dk  then  norm(q)  is     det (   +      +   ) = a2 + b2 + c2 + d2−  +      −             since  det(q1) det(q2) = det(q1q2)  we  can  also  write  the  “complex  two  square  identity”  as  a  real  four  square  identity  .  which  turns  out  to  be      (a21 + b21 + c21+ d21)( a22 + b22 + c22+ d22) =                                                        (a1a2 – b1b2–  c1c2 –d1d2)2                                                     +(a1b2 – b1a2 + c1d2 – d1c2)2                                                  +(a1c2 - b1d2 +c1a2 + d1b2)2   +(a1d2 + b1c2 –c1b2+ d1a2)2 Remarkably ,  the  four  square  identity  was  discovered  by  Euler  in  1748,   nearly  100  years  before  the  discovery of  quaternion.  Euler  hoped  to  use  it  to  prove  that  every  natural number  is  the  sum  of  four  squares,  by  proving  also  that  every  prime  is  the  sum  of  four  squares.   This  was  first  proved  by  Lagrange  in  1770.  We  can  now  give  a  simpler  proof  with  the  help  of  quaternion.  ℤ[  ,   ,  ]:        From  now  on  we  write  the  quaternion  1  simply  as  1  and  omit  it  altogether  as  a  item  in  a  product  .  thus  the  typical  quaternion  will  be  written.      q= a + bi + cj + dk,    where     a , b , c , d ∈ℝ.   Which  of  these  objects  should  be  regarded  as  “integer”?                  One ‘s    first  thought  is  that    ℤ[  ,   ,  ] = { a + bi + cj + dk:  a , b , c , d ∈ℤ }     Should  be  the  “quaternion  integer”  analogous  to  the  Gaussian  integers  ℤ[i].  Sum  ,  difference  and  product  of  members ℤ[  ,   ,  ] are   again  members  of  ℤ[  ,   ,  ] and       Norm(a + bi + cj + dk) = a2 + b2 + c2 + d2 Is  an  ordinary  integer,  which  we  can  use  to  find  “primes”  in  ℤ [  ,   ,  ].  EXAMPLE:                   2 + i + j + k  is   a  prime  of ℤ[  ,   ,  ]. SOLUTION;      This  is  so  because  norm  ,     (2 + i + j + k) = 22 + 12 + 12 + 12                                                 =7           Which  is  a  prime  in  ℤ , hence  2 + i + j + k is  not  the  product  of  members  of ℤ[  ,   ,  ]with  smaller  norm.  HURWITZ  INTEGER:    The  quaternion  in ℤ [1 + + +  ,i ,j ,k ]  are  called  the  Hurwitz   integers. CONCLUSION:  In this   paper     discussed  about  number  theory  using  Pell  equation.  Also  we   have discussed      Brahmagupta   composition   rule , the quaternion  units , the  Hurwitz  integers , conjugates , a prime  divisor  property  ,  and   four  square  theorem.   As  well  as  the well – known  applications  to  many    others  areas  of  mathematics. Medical   science  is  not  an  exact  science  in  which  processes  can  be  easily  analysed  and  modelled.        These   are   all   the   field   of   current   research  in   number   theory. BIBLIOGRAPHY: [1] John stillwell (2003),  “Elements Of Number Theory” springer – verlag, New York.  [2] Redmond .D  (1996) , Number  theory  marcel  Dekker Inc., New York.  [3] Yan, S.Y. (2000) .Number  theory  for  computing.  Springer – verlag,Berlin.    

A STUDY ON FOUR SQUARE THEOREM

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A STUDY ON FOUR SQUARE THEOREM

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