# Mathematics – Number Theory

Number theory is a branch of pure mathematics which is very vast; it is actually deals with whole numbers or rational numbers which are commonly referred to as, fractions. The elementary number theory elaborates on the divisibility of integers, the division or algorithm. Nevertheless the number theory is intertwined to other branches of mathematics for instance every natural number is the sum of four integer squares and this posits that X2+ Y2+ Z2 +W2 = n (Rusin, 2006 p. 1). The theory also considers elementary properties of primes that is, factorization theorem and the infinitude of primes.

In the geometric number theory incorporates all forms of geometry; classical geometry of numbers due to minkowski begins with statements of Euclidian (Chakraborty et al, 2008, pp. 597-611). By extension this becomes the study of quadratic forms on lattices, and thus a method of investigating regular packaging of spheres (Rusin, 2006 p, 1). Number theory can also be used to investigate algebraic geometry that is by way of studying the varieties like algebraic curves and surfaces i. e. cubes and three dimensional surfaces. Numerical analysis is done to make mathematics a more applicable than theoretical field.

Galois theory is for instance useful in field extensions; its analytical tools are useful in data collection and compiling (Rusin, 2006 p. 1). Other applications of the theory are general such as programming in the technological domain and analysis of the tools of arithmetic. The proposed theories form the basis of a better understanding of the mathematical concepts. The mathematical functions are studied in view of the need to use the subject to solve emerging problems which may be arithmetic (Chakraborty et al, 2008, pp. 597-611).

Unique Factorization Theorem The commutative Mobius monoid satisfies a unique factorization theorem because it has arithmetic features same to ones of multiplicative semi group with positive integers (Alaca et al, 2008, pp. 677-689). The bicyclic semi group together with the multiplicative analogue of the group is given prominence because of its centrality in understanding the unique factorization concept as outlined by Schwab & Haukken (2006, p. 549)

Fundamental theorem of arithmetic is key to the understanding of unique factorization theorem in commutative Mobious monoids, nevertheless as an application the theory is defined on arbitrary commutative Mobious monoid terms with regard to general arithmetical functions. Schwab & Haukken (2006, p. 549) proposes that the Mobiuos monoid in the event that S is a combinatorial inverse monoid poset E(S) of idempotent is locally finite (Chakraborty et al, 2008, pp. 597-611). The minimal standard division category of S brings about the general program to enhance the study of the combinatorial properties (Schwab & Haukken 2006, p. 549).

Apparently the R-class R1 of the bicyclic semi group is commutative Mobious Monoid isomorphic to the additive semi group of non negative integers. Some of the major characteristics of the inverse semi groups include; 1. (s? 1) ? 1 = s for every s ? S, 2. (s1s2 · · · sn) ? 1 = s? 1…. s 2-1 for all s1, s2, . . . , sn ? S, s? 1s and ss? 1 are idempotents for any s ? S, 3. if e ? E(S), then e? 1 = e and e? 1e = ee? 1 = e, 4 eS ? fS = efS and Se ? Sf = Sef for any e, f ? E(S), 5 (E(S), ? ) is a meet semilattice (Schwab & Haukken 2006, p. 549).

The arithmetic on the R- Class containing identity of a combinatorial bisimple inverse monoid is exemplified by; the formula below. Let S be a bisimple inverse monoid. Therefore the R-Class R1 containing the identity be; R1= {s E S|ss-1=1}. The canceling monoid satisfies the condition for the s, t E R1 and there is X E R1 so that R1s nR1t =R1x (Schwab & Haukken 2006, p. 551). To prove the above, we let s ,t E R1and st(st) -1 =stt -1s-1=ss-1=1 therefore st E R1 further, 1E R1. thus R1 is a monoid. R1 is the right cancellative because if s, t, u E R1 then su=tu conclusively, Suu -1=tuu-1 and thus s=t.

(Chakraborty et al, 2008, pp. 597-611). Anti-Hasee Principle Theorem This theory basically deals with curves as arithmetic solutions, these curves are moduli spaces of elliptic Q-curves for instance C (N, p) (Q) parameterizes elliptic curves defined by Q (vp) that are cyclically N-isogenies to their Galois conjugates. The axiomatic version of the curve that; H1 (Q/Q, (o)) = Hom (Gal (Q/Q), ± 1) = Qx/Qx2. Let C/q be an algebraic curve. Suppose there is a Q-rational involution on C so that {P E C (Q) |I (P) =P}= o (Clark, 2006, p. 628).

The sequence of curves XD+ is the example per excellence of a naturally occurring family of curves with points everywhere locally besides works of Jordan outline that the modular interpretation of XD+ is in many respects more natural than that of XD . When N=14 and P=17 ( mod 56),computations suggest that there are always points everywhere locally. Although there are no usual Q-rational points here, the first 101 primes in the order can get an elliptic curve (Clark, 2006, p. 628). Galois Group Realizations To distinguish explicit groups as Galois groups over Q is quite a task and is rather limited.

The study of the Galois groups principle is however very vital in solving a homogeneous group of linear equations (Farag, 2008, pp. 653-662). In that regard the solvable transitive group of degree 18 and order 2. 34, as a Galois group over Q, the group has two no conjugate Gassman equivalent mini groups of the given index i. e. 18 (Salvador-Villa 2005, p 641). The first theorem proposes that; if you let L/K be a Galois extension of number fields given the Galois group Gal (L/K) = S. U being the abelian group of exponent n where S acts. Then G = U ?

S. The proof is that the extension M1/K is a normal extension because it conjugates in nvs (li) over K, s E S, i? i? h belong to M1. If µ: Gal (M1/K)>S the restriction is to L and U1 = ker (µ), then the exact sequence becomes 1 > U1 _ Gal(M1/L) > Gal(M1/K) ? > S _ Gal (L/K) > 1 (Salvador-Villa 2005, p 645). Another example proposes that if you let p be ? 3 then a prime number and (g) be the cyclic group of even order n with gcd (n, p) = 1. P will be = (x) ? (a, b) the only group of order p4 such that exp (P) = p, o (Z (P)) = p2, [P: Frat (P)] = p3.

The group (a, b) is a nonabelian maximal subgroup of P and [a, b] Z (_a, b_) we consider the group G = P <g of order np4 (Salvador-Villa 2005, p 643). The linear equations follow the above exemplified formula to be solved however in certain occasions there are many other formulas that are proposed to be systematic depending on the complexity with which the sum needs to be tackled. Some of the equations can use a generalized approach while others can assume a specific approach towards getting the solution (Salvador-Villa 2005, p 650). Theory of Diphonantine Approximations

This is a number theory of a problem of Erdos, Szusz and Turan regarding Diphonantine approximations. This number theory problem was elevated by Erdos, Szusz and Turan. There are sets of ? ?] 0, 1[which denotes the S (N, A, c). This is expressed by A>0, c>1 and each integers N? 2. The coprime integers in this theory are a and q (Boca, 2007, p. 706). Therefore the derived formula for this theory is shown in equation 1 in the appendix. There is a problem that exists in the limit f (A, c) when measuring Lebesgue | S (N, A, c)| in this measure N = ?. When Ac ? 1, then the limit f (A, c) is proven to exist.

This is done using the formula in the equation 2 in the appendix (Boca, 2007, p. 691). In this formula A ? ? when ? = (1+ (v1-v4A2))/ 2A also A ? ? when ? =1. There is existence of f (a, c) in the values of c> 1 and A> 0. This can be shown in the using the transformation which is the bijective area-preserving expressed by equation 3 in the appendix. In this formula the inverse of T can be given by STS where S(x, y) = (y, x). The first theorem of this theory is shown by equation 4 in the appendix. The formula 4 in the appendix is used for finding the existence of N = ?.

This can be done when c> 1 and A> 0, where the limit is | S (N, A, c)| (Boca, 2007, p. 693). In calculating f (A, c) when Ac ? 1, the formula of the theorem one will be used. The 0< A< 1/c is used in comparing the f (A, c) and theorem one. Dc = D+c = TD – c when c ? 2. This formula is used when 1 ? c< 2 (Boca, 2007, p. 706). Sum of Squares and Sums of Triangular Numbers induced by partitions of 8 In this theory we let rk (n) and tk (n) represent the number of representations of an integer n as a summation of k triangular numerals, respectively (Masri, 2008, pp. 613-626).

The aim is to provide evidence that t8 (n) = 1/210 * 32 [r8 (8n + 8) _ 16r8 (2n + 2)]. Therefore the analysis of the sequence t8 (n) is brought down to the study of subsequences of r8 (n). In this case, an additional 21 analogous findings pertaining to summations of squares and summations of triangular numerals induced by partitions of 8 (Baruah et al, 2008, p. 525). What is meant is that rk (n) represents the number in integers of the equation x12 + … + xk2 = n (Alaca et al, 2008, pp. 677-689). Supposing that 1? k ? 7. What would follow is that for any non-negative integer n, we would have rk (8n + k) = cktk (n) (Masri, 2008, pp.

613-626). Therefore the examination of tk (n) for 1 ? k ? 7 is eventually reduced to the examination of the subsequence rk (8n + k) of rk (n). In a bid to state the findings, let ? = (? 1, …, ? m) be a partition of k. What this means is that ? 1, …, ? m are integers which meet the condition ? 1 ? …? ?m ? 1 and ? 1 + … + ? m = k. Therefore fore any integer which is not negative n, r? (n) is defined to be the number of solutions in term s of integers of ? 1×21 + … + ? mx2m = n (Baruah et al, 2008, p. 526). According to Baruah et al (2008, p. 526), to provide a proof of the above theorem, we let ? = (? 1,…, ?

m) form part of 8 (Masri, 2008, pp. 613-626). We further let a? (n) represent number of integral solutions in ? 1×21 + … + ? mx2m = n. In this case, x1, …, xm, are not all even numbers. We further let b? (n) represent the number of solutions in integers which give an odd result for x1,…, xm.. Then for any non- negative integer n, we would have a? (8n) = C? /2m * b? (8n). (See appendices for C? formula). Wilson’s Theorem With regards to we indicate the theorem pertaining to the primes congruent to 1 and 3 modulo in a respective order. For any integer n, clearly limq>1 [n]q = n, so it is said that [n]q is a q- analogue of the integer n.

Let us assume that a ? b then [a]q = (1- qa/1-q). In this case the above congruence is given consideration over the polynomial ring ? [q] in the variable q having integral of coefficients (Chapman & Pan, 2008, p. 539). At the same time, q-analogues of particular arithmetical congruence have been examined. In this case we consider p a prime. The popular Wilson’s theorem states that (p – 1)! ? -1. In this case let us use a q-analogue of Wilson’s theorem for a prime p? 3. In the theory, Mordell gave a proof that should p> 3 is a prime and p ? 3 then (p – 1/2)!

? (-1) h(-p) +1/2 (mod p) in which case h(-p) represents the number of the quadratic field Q(v-p). This would now enable us to determine a q-analogue of mode p. The case p? 1 (mod4) is quite complex. By use of definition, for any prime to p, (a/p) = 1 or -1 depending on whether a represents a quadratic residue modulo p (Chapman & Pan, 2008, p. 539). Assume that ? p > 1 and that h(p) form the integral unit and the class number of Q (vp) in a respective manner (Chapman & Pan, 2008, p. 539). To prove the above theorem we make an assumption that p > 3 is a prime that p? 3 (mod 4).

(See appendices for formula). An observation is made thus [p]q = 1- qp/1- q. (See appendices). In such a case ? = ? 2? і/p. At the same time it is evident that ? Ѕ : ? > ? Ѕ represents an automorphism over Q(? ) as long as p | Ѕ (Chapman & Pan, 2008, p. 539). Siegel Modular Forms The ring of genus three Siegel modular forms has been given by Runge as a quotient ring, R3/ {J (3)} in which case R3 represents the genus three ring of code polynomials. J(3) on the other hand is a representation of the weighted numerators of ? 8 and d +16 codes (Oura et al, 2008, p. 563).

It needs to be stated that an important ring homomorphism exists. This is Th2 : Rg > Mg, from code polynomials to Siegel modular forms pertaining to genus g. The ring of code polynomial is described as the Hg invariant subring. What happens is that a right action of Spg (R) regarding such functions. These are the functions of ? : ? g > С meeting ? | k? = ?. Still, what needs to be understood is that the map Th2 is described by sending the factors Fa to the second order theta constant. In genus three, Runge demonstrated that the Th2 kernel brought about by a degree 16 polynomial J(3) was related with the Schottky form.

This follows that what Runge did was to give a representation of the complex ring M3 as a quotient ring R3/ {J(3)}. An attempt was made towards the same goal in genus four by determining R(4)0, a polynomial relation of degree amidst 24 second order constants of theta. This feat was made possible by the fact that the generating function for R4 had already been computed, the dimension of M124 was known alongside the fact that Siegel modular forms established by the Fourier coefficient on root latticed they had (Oura et al, 2008, p. 564). Conclusion

In conclusion assuming that L is an integral rank g lattice with ? = exp (L*/ L). The map O*L : Mg >M1 (? 0(? )) is considered a graded ring homomorphism which performs a multiplication of weights by g besides taking cusp forms (Chakraborty et al, 2008, pp. 597-611). What emerges from here is that is easy to deal with a series of one variable power. In general the strategy is to find a replacement for the computation of Fourier coefficients of ? which is done by the specialization O*L? at any time when possible (Oura et al, 2008, p. 564).

Usually these mathematical computations are undertaken by the use of such techniques as C++, Fermat, Mathematica, GAP as well as Magma among others. In a nutshell it is important to appreciate that there are several theories in mathematics. Usually, proving all of them can at times be an elaborate and complicated process but usually very interesting. References Alaca, A. , Alaca, S and Williams K. S. (2008). Berndt’s Curious Formula. International Journal of Number Theory. 4 (4), 677-689. World Scientific Publishing Company. Boca, F. P. (2007). A Problem of Erdos, Szusz and Turan concerning Diphonantine approximations.

International Journal of Number Theory. 4 (4), 691-708. Baruah, N. D. , Cooper, S. , and Hirschhorn, M. (2008). Sum of Squares and Sums of Triangular Numbers Induced by partitions of 8. International Journal of Number Theory Vol. 4 (4), 525-438. World Scientific Publishing Company. Chakraborty, K. , Luca, F. , and Mukhopadhyay, A. (2008). Exponents of Class Groups of Real Quadratic Fields. International Journal of Number Theory 4 (4), 597-611. World Scientific Publishing Company. Chapman, R. , & Pan, H. (2008). q-Analogues of Wilson’s Theorem. International Journal of Number Theory 4 (4), 539-547.

World Scientific Publishing Company Clark, P. L. (2006) An “Anti-Hasee Principle” For Prime Twists. International Journal of Number Theory. 4 (4), 549-561. World Scientific Publishing Company. Retrieved December 23, 2008 from http://www. math. niu. edu/~rusin/known-math/index/11-XX. html Clark, P. L. (2003) Rational Points on Atkin-Lehner quotients of Shimura curves, Ph. D. thesis, Harvard University. Retrieved December 23, 2008 from http://www. math. niu. edu/~rusin/known-math/index/11-XX. html Farag, H. M. (2008). Dirichlet Truncations of the Riemann Zeta Function in the Critical Strip Possess Zeros Near Every Vertical Line.

International Journal of Number Theory 4 (4), 653-662. World Scientific Publishing Company. Masri, N. (2008). Infinite Families of Sums of Formulas for Sums of Integer Squares. International Journal of Number Theory 4 (4), 613-626. World Scientific Publishing Company. Oura, M. , Poor, C. , and Yuen, D. S. (2008). Towards the Siegel Ring in Genus Four. International Journal of Number Theory 4 (4), 563-586. World Scientific Publishing Company.

Rusin, D. (2006) Number Theory: The Mathematics Atlas. Retrieved December 23, 2008 from ttp//hwww. math-atlas. org/welcome. html Salvador-Villa, G. D. (2005), Explicit Galois Group Realizations. International Journal of Number Theory. 4 (4), 639–652 World Scientific Publishing Company. Retrieved December 23, 2008 http://www. math. niu. edu/~rusin/known-math/index/11-XX. html Schwab, E. D. , & Haukkanen, P. (2006). A Unique Factorization in Commutative Mobius Monoids. International Journal of Number Theory. 4 (4), pp. 549-561. World Scientific Publishing Company. Retrieved December 23, 2008 from http://www. math. niu. edu/~rusin/known-math/index/11-XX. html

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