Ramanujan s Famous Partition Congruences

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1 Ope Sciece Joural of Mathematics ad Applicatio 6; 4(): - ISSN:8-494 (Prit); ISSN:8-494 (Olie) Ramauja s Famous Partitio Cogrueces Md Fazlee Hossai, Nil Rata Bhattacharjee, Sabuj Das Departmet of Mathematics, Uiversity of Chittagog, Chittagog, Bagladesh Departmet of Mathematics, Raoza Uiversity College, Chittagog, Bagladesh address sabujdasctg@gmailcom (S Das) To cite this article Md Fazlee Hossai, Nil Rata Bhattacharjee, Sabuj Das Ramauja s Famous Partitio Cogrueces Ope Sciece Joural of Mathematics ad Applicatio Vol 4, No, 6, pp - Received: August 8, 6; Accepted: August 6, 6; Published: September, 6 Abstract I 4, firstly Leohard Euler iveted the geeratig fuctio for P(), where P() is the umber of partitios of [P() is defied to be ] Sriivasa Ramauja was bor o December 88 I 96, S Ramauja iveted the geeratig fuctio for P() ( d time) Godfrey Harold Hardy said Sriivasa Ramauja was the first, ad up to ow the oly, Mathematicia to discover ay such properties of P() MacMaho established a table of P() for the first values of, ad Ramauja observed that the table idicated certai simple cogrueces properties of P() I 96, S Ramauja quoted his famous partitio cogrucecs I particular, the umbers of the partitios of umbers m+4, m+, ad m+6 are divisible by,, ad respectively Now this paper shows how to prove the Ramauja s famous partitios cogrueces modulo,, ad respectively Keywords Cogrueces, Eumeratig, Modulo, Residues, Ramauja s Lost Notebook Itroductio I this paper we give some related defiitios of P( ), Euler s product, Jacobi s product, Triagular umber, λ We discuss Petagoal umber, the geeratig fuctios for λ, λ, ad λ, λ ad λ respectively ad prove the Ramauja s famous partitio 4 mod P m + mod cogrueces: P( m + ), ad P( m + 6) ( mod) with the help of Euler s product, Jacobi s product ad particular cogrueces [Ramauja s Lost Notebook (96)] { } Some Related Defiitios ( ) ( ) ( ) {( ) ( ) ( ) } mod { } 4 ( mod ), ad ( mod ) P( ) : The umber of partitios of like 4, +, +, ++, +++ P ( 4) = Euler s product:

2 8 Md Fazlee Hossai et al: Ramauja s Famous Partitio Cogrueces Jacobi s product: ( )( )( ) = = ( ) ( ) = - λ {( )( )( ) } = + + = ( -) ( + ) 6 : The umber of partitios of as the sum of itegers, which are ot multiples of λ : The umber of partitios of as the sum of itegers, which are ot multiples of 49 λ : The umber of partitios of as the sum of itegers, which are ot multiples of Triagular umber: A iteger that ca be represeted by a triagular array of dots like,, 6,,,, 8, etc or shortly m m + itegers) That is, Triagular umbers 6 8 etc ; m N (the set of atural ( ± ) m m Petagoal umbers: The umbers = are called the petagoal umbers like,,,, where m N (the set of atural itegers) If we cosider a regular petago marked by dots, so as to obtai successively petagos with, 4,, dots o each side, the the total umber of The Geeratig Fuctio for λ ( ) dots is = = + (-) ( ) = - ( ) m m ( + ) ( + ) ad is show by followig figure, Petagoal umbers : If m = ; m = ; m = ; m = 4 ; m = ; m = ( m ) = = = = = Figure The petagoal umbers like,,,, [S Ramauja (96)] Is Give by ( ) ( ) ( ) { } = P = = P P P + P + = = = = = = P P 6 P + P 6 + P 6 ] Where, 6,, 6, are umbers of the form ( m ) ( m ) iteger Therefore, ad ( m + ) ( m + ), such that m is ay positive

3 Ope Sciece Joural of Mathematics ad Applicatio 6; 4(): - 9 ( ) ( ) ( ) ( m ) ( m ) ( m + ) ( m + ) m m P P P λ m = = m= = = + + ] =, m where, λ m m m m + m + = P + P + P m= m= So the coefficiet λ of i the above epasio is the umber of partitios of as the sum of itegers, which are ot multiples of It shows that the coefficiet of i the above epasio 4 9 P 4, is a multiple of ad we follow that P, P, ( mod), ie, ( 4) ( mod) P m +, where m is a oegative iteger Now it eeds to be proved P m + 4 mod Theorem : Proof: We get; { } { } { } 4 = = ( ) -- + µ µ + + ν ν + = ν + [By Euler ad µ = ν = Jacobi s idetities] () = + ( ) ( ν ) µ = ν = k, where k = k ( µ, ν ) = + µ ( µ + ) + ν ( ν + ) We cosider the ide of is divisible by, ad the ide ( ) ( ) But ( ) ( ) k = + µ ( µ + ) + ν ( ν + ) k = + + Now, µ + + ν + = µ + 4µ + + 4ν + 4ν + = 8k µ is also divisible by We have; ( µ + ), or ( mod) ad ( ν + ), or 4( mod), we see by eumeratig the possible cases, if µ 4( mod) ad ν ( mod), the the sum of residues from above two cogrueces is, cosequetly the every term is modulo Hece if the ide k is divisible by, the coefficiet (ν + ) i () is also divisible by, ad therefore the coefficiet of { } 4 m + i ( ) is divisible by ( ) ( ) m + i ( ) { } ( ) ( ) 4 {( ) ( ) } Hece the coefficiet of = multiple of Sice, = ( ) ( ) ( ) ( ) m + Hece the coefficiet of i = + P( ) is divisible by = Thus P( m + 4) ( mod) Hece the Theorem Eample : We get; p p p 4 =, 9 =, 4 =, i, e, p( 4) = ( mod ), p( 9) = ( mod ), p( 4) = ( mod ), We ca coclude that P( m + 4) ( mod), where m is a o-egative iteger ( ) ( ) ( ) ( ) ( ) ( ) { } ( mod) is a 4 The Geeratig Fuctio for λ [S Ramauja (96)] Is Give By ( )( )( ) { } = 4 p ( )( )( )( ) = = P P P + P + = = = =

4 Md Fazlee Hossai et al: Ramauja s Famous Partitio Cogrueces = P ( ) P( ) P( ) + P( 4 ) + P( 4 ) ], = where,, 4, 4,, are umbers of the form m m 4 ad m + m + 4 such that m is ay positive iteger Therefore, ( )( )( ) 4 ( )( )( ) ( m ) ( m 4) ( m + ) ( m + 4 ) m m P P P λ m = = m= = = + + ] =, m where, λ m m 4 m m + m + 4 = P + P + P m= m= λ of So the coefficiet i the above epasio is the umber of partitios of as the sum of itegers, which are ot multiples of 49, It shows that the coefficiet of i the above epasio is a multiple of We follow that ( mod), ie, ( ) ( mod) P m +, where m is a oegative iteger P, P, P ( 9), Theorem [Ramauja (96)]: P( m + ) ( mod) Proof: We get; { } { } { } 6 = 4 µ µ ν ν = + + µ = ν = Jacobi s idetity] + µ µ + + ν ν + = µ + ν ( µ ) ( ν ) [By µ = ν = = k ( µ ) ( ν ) () µ = ν = k = Where k = + µ ( µ + ) + ν ( ν + ) 8k = Agai, ( µ ) ( ν ) ( µ ν ) ( µ ν ) = = 8k 4, which ca be divisible by oly if the ide k is divisible by Now we have; ( ),, or 4( mod) ( ν + ),, or 4( mod) µ + ad We ca see by eumeratig the possible cases if mod ν mod, the the sum of residues µ ad from above cogrueces is Therefore the every term is modulo Hece, if the ide i () is a multiple of, the µ + ν + is also multiple of, ad { } coefficiet such that m is ay positive iteger Therefore, therefore the coefficiet of {( ) ( ) ( ) } 6, m + is a multiple of Hece we ca easily verify that the coefficiet of ( ) 4 { } ( ) ( 4 ) 6 ( ) = is a multiple of So the coefficiet of ( ) ( ) { } m + i P m + i i is a = + = multiple of P m + mod Hece the Theorem Thus Eample : We get; p =, p =, i, e, p = ( mod ), p = ( mod ), We ca coclude that P( m + ) ( mod), where m is a o-egative iteger The Geeratig Fuctio for λ [S Ramauja (96)] Is Give by 4 6 ( )( )( ) 4 ( )( )( )( ) { } 4 6 = P = = P P P + P + = = = = = = P ( ) P ( ) P ( ) + P ( ) + P ( ) ], where 6, 4, 6, 8, are umbers of the form ( m ) ( m ) ( ad m + ) ( m + ),

5 Ope Sciece Joural of Mathematics ad Applicatio 6; 4(): ( )( )( ) 4 ( )( )( )( ) m m m + m + = P + P + P = m= m= m m = λ m Where, λ = m m m m + m + = P + P + P m= m= λ of So the coefficiet multiples of, It shows that the coefficiet of ( mod), ie, ( 6) ( mod) i the above epasio is the umber of partitios of as the sum of itegers, which are ot i the above epasio is a multiple of We follow that ( 6) P m +, where m is a o-egative iteger Now we prove the Theorem P m + 6 mod Theorem : { } Proof: We get; ( ) { } { } { } { } = ( + ) ( + ) m ( m + ) ( + ) m m m m r r m + m + m + r = ( ) ( m + )( m + )( m + ) 8 m = m = m = r = [By Euler ad Jacobi s idetities] Where, m + m + m + r = ( ) ( m + )( m + )( m + ) 8 m = m = m = r = = ( + ) ( + ) m ( m + ) ( + ) m m m m r r P, P ( ), P ( 8) m + m + m + r k ( m )( m )( m ) () m = m = m = r = ( + ) ( + ) m ( m + ) ( + ) m m m m r r k = k = m + m + m + m + m + m + r + r + ( ) 8k 4 m m m m m m r r 4 = We have; ( m + ) + ( m + ) + ( m + ) + ( r + ) ( ) = m m m m m m r r = 8k r, which ca be divisible by oly if the ide k is divisible by Now, ( m ) +,,,4, or 9( mod) ( m ) +,,,4, or 9( mod) ( m ) +,,,4, or 9( mod) ad is a multiple of Fially, the coefficiet of,,,4, or 9( mod) r +, We ca see by eumeratig the possible cases if m mod, m ( mod) ad m ( mod), r ( mod), the the sum of residues from above cogrueces is Cosequetly the every term is modulo Hece if the ide k i () is divisible by, the m + m + m + is also divisible { } coefficiet by, ad therefore the coefficiet of {( ) ( ) } is divisible by Hece we ca easily verify that the coefficiet of = {( ) ( ) } m+ i { } m+ m+ i i

6 Md Fazlee Hossai et al: Ramauja s Famous Partitio Cogrueces = + = 6 ( ) P ( ) is a multiple of Thus P( m + 6) ( mod) Hece the theorem Eample : We get; p( 6) =, p( ) = 9, i, e, p ( 6) = ( mod ), p ( ) = 9 ( mod ), We ca coclude that P( m + 6) ( mod), where m is a o-egative iteger 6 Coclusio I this paper we have defied the Triagular umber, Petagoal umbers with the help of figures For ay oegative iteger of m we have proved the theorem ad have verified the Theorem with m =,, We have established the Theorem ad have verified the Theorem with m=,, Fially we have proved the Theorem by algebraic method ad have verified the Theorem with m=,, respectively Refereces [] G E Adrews, A Itroductio to Ramauja s Lost Notebook, Amer Math Mothly, 86, (99), pp 89-8 [] G E Adrews, A Itroductio to Ramauja s Lost Notebook ad other upublishedpapers, Norosa Publishig House, New Delhi, (988), pp- [] B C Berdt, Ramauja s Notebook, Part III, Spriger- Verlag, New York, (99)pp - [4] GH Hardy, ad E M Wright, Itroductio to the Theory of Numbers, 4 th Editio, Oford, Claredo Press, (96) [] MacMho, Combiatory aalysis, A Arbor, Michiga: Uiversity of Michiga Library () [6] R A Raki, Ramauja s mauscripts ad otebooks II, Bull Lodo Math Soc (989), -6, reprited i [, pp 9-4] [] S Ramauja, Ramauja s had writig letter to Hardy ad sheets, (96) [8] S Ramauja, Highly composite umbers, Proc Lodo Math Soc 4 (9), 4-4 [9] S Ramauja, Notebooks ( volumes), Tata Istitute of Fudametal Research, Bombay, 9 [] S Ramauja, The Lost Notebook ad Other upublished Papers, Narosa, New Delhi, 988