Read Stillwell

I wish to draw attention to Stillwell's article "Number Theory as a Core Mathematical Dicipline". Stillwell says: "My suggestion is that mathematics, from kindergarten onwards, should be built around a core that is interesting at all levels, capable of unlimited development, and strongly connected to all parts of mathematics. My paper attempts to show that number theory meets these requirements, and that it is natural to build modern mathematics around such a core." The rest of the paper is a wonderful display of beautiful number theory with deep connections with all major areas of mathematics. Naturally there are connections with algebra, and "this is not surprising, because most basic commutative algebra is derived from Gauss's Disquisitiones Arithmeticae via Dirichlet and Dedekind". For example, a "wonderful constellation of results comes from forming the product of elements in an abelian group in two ways", for instance Fermat's little theorem: a^(p-1)=1 mod p when gcd(a,p)=1. This is because the list a,2a,...,(p-1)a mod p must be a permutation of 1,2,...,(p-1) mod p since all elements are nonzero and unequal (since a is invertible), so a*2a*...*(p-1)a=1*2*...*(p-1) mod p, i.e. a^(p-1)*1*2*...*(p-1)=1*2*...*(p-1) mod p, i.e. a^(p-1)=1 mod p. There are also connections with complex numbers. Since (a+bi)(c+di)=(ac-bd)+(ad+bc)i, the multiplicative property o absolute value reads (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2, which is a classical number theoretic identity apparently know to Diophantus when he said things like "65 is naturally divided into two squares in two ways, namely into 7^2+4^2 and 8^2+1^2, which is due to the fact that 65 is the product of 13 and 5, each of which is the sum of two squares". Now, 13 and 5 in turn are sums of two squares "because" they are primes of the form 4n+1, which is a theorem most easily proved 1,0) with rational slope t, so we find them t^2)/(1+t^2) and y=2t/(1+t^2). This rationalisation of the circle also pays of in calculus x^2). Indeed, Bernoulli explicitly credited Diophantus for the substitution he used to turn the integrand for the arc length of a circle into 1/(1+t^2) from where the infinite series of pi follows x^4) cannot be rationalised. Assume it can, sqrt(1-x^4)=y, square both sides and multiply up denominators to get Z^4-X^4=Y^2. the impossibility of this when X,Y,Z are integers was essentially proved by Fermat, by infinite descent, and polynomials behave sufficiently like integers for us to be able to mimic his proof in the polynomial case.