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Assume every large integer is the sum of three primes

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Homework 5 (circle method). Due by May, 2.

1. Assume every large integer is the sum of three primes. Prove every large even integer is the sum of two primes. Conversely, show if every large even integer is the sum of two primes, every large integer is the sum of three primes.

2. Show that

r1,s(N) =

N −1 s−1

= Ns−1

(s−1)! +O(Ns−2).

Show that this bound is consistent with the Hardy-Littlewood asymptotic formula for k = 1.

3. Let f(x)be a polynomial of degree k ≥2with integral coefficients and let

Sf(q, a) =

q

X

r=1

e(af(r) q ).

Frove that if(q, r) = 1, then

Sf(qr, ar+bq) =Sf(q, a)Sf(r, b).

1

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