primes of the form aq2+1
TRANSCRIPT
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Primes of the form aq2 + 1
Kaisa MatomÀki
Department of Mathematics
Royal Holloway, University of London
March 6, 2008
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Outline
1 Approaches to primes of the form n2 + 1
2 Primes of the form aq2 + 1
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Primes of the Form n2 + 1
A long-standing conjecture:
Conjecture
There are inïżœnitely many primes of the form n2 + 1.
The ïżœrst twelve instances of such primes are
2 = 12 + 1 101 = 102 + 1 577 = 242 + 1
5 = 22 + 1 197 = 142 + 1 677 = 262 + 1
17 = 42 + 1 257 = 162 + 1 1297 = 362 + 1
37 = 62 + 1 401 = 202 + 1 1601 = 402 + 1
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Primes Represented by Polynomials
The case of primes of the form n2 + 1 = f (n) is a special case of amore general conjecture.
Conjecture
Any reasonable polynomial f (n) â Z[x ] takes prime values inïżœnitely
often.
The linear case f (n) = an + b (where reasonable meansgcd(a, b) = 1) was proved by Dirichlet in 1837.
No instance of the conjecture in higher degree is known.
In 1922, Hardy and Littlewood gave a conjectural asymptoticformula for the number of primes of the form f (n).
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Prime factors of n2 + 1
Theorem (Iwaniec 1978)
There are inïżœnitely many integers n such that n2 + 1 has at most
two prime factors.
Write P(m) for the greatest prime factor of m. Then of course
m â P ââ P(m) = m.
The previous theorem implies P(n2 + 1) > n inïżœnitely often.
Theorem (Deshouillers and Iwaniec 1982)
There are inïżœnitely many integers n such that P(n2 + 1) > n6/5.
In the proof the authors use their deep results on averages ofKloosterman sums.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Approximations with Polynomials in Two Variables
Theorem (Harman and Lewis 2001)
There exists inïżœnitely many primes p with
p = n2 + m2, m, n â Z,m < p0.119.
The proof uses a generalization of Harman's sieve method toGaussian integers (complex numbers m + ni with m, n â Z).
Theorem (Friedlander and Iwaniec 1998)
There are inïżœnitely many primes of the form n2 + m4.
The proof is almost one hundred pages long!
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Approximations Using Fractional Parts
For x â R, write {x} for the fractional part of x .
So {x} â [0, 1) and x â {x} â Z.If 0 < {x1/2} < xâ1/2, then for some integer n,
n < x1/2 < n + xâ1/2 =â n2 < x < n2 + 2.
Thus if we could show that {p1/2} < pâ1/2 inïżœnitely often,our conjecture would follow. But we only know
Theorem (Balog 1983, Harman 1983)
For any Δ > 0, there are inïżœnitely many solutions to
{p1/2} < pâ1/4+Δ.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Approximating with aq2 + 1
Still another way to approach primes of the form n2 + 1 is toconsider primes of the form p = aq2 + 1 with a as small aspossible.
The case a = 1 is of course the conjecture itself.
Baier and Zhao 2006: a †p5/9+Δ for any Δ > 0.
Theorem (MatomÀki)
Let Δ > 0. There are inïżœnitely many primes of the form
p = aq2 + 1, where a †p1/2+Δ and q is a prime.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Reïżœning the Task
Let X be a large number. We write, for q2 ⌠Q = X 1/2âΔ,
A(q) = {aq2+1 | aq2+1 ⌠X} = {n ⌠X | n ⥠1 (mod q2)}.
We need that for inïżœnitely many q holdsâ
pâA(q) 1 > 0.
No reason for the residue class 1 (mod q2) to be special amongthose coprime to q2 and thus deïżœning
B(q) = {n ⌠X | gcd(n, q2) = 1}
we would expect thatâpâA(q)
1 =1
Ï(q2)
âpâB(q)
1 + smaller errorPNT=
X (1 + o(1))
Ï(q2) logX.
This holds assuming the generalized Riemann hypothesis.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Sums over Primes
Sums over primes diïżœcult to handle.
But, it is possible to decompose them into easier sums.
We want to splitâ
pâA(q) 1 â1
Ï(q2)
âpâB(q) into showing
that for type I sumsâmnâA(q)mâŒM
am =1
Ï(q2)
âmnâB(q)mâŒM
am + error
and type II sumsâmnâA(q)mâŒM
ambn =1
Ï(q2)
âmnâB(q)mâŒM
ambn + error ,
where M lays in certain ranges.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Type I Sums
For a set E â N, we write Ed = {m | dm â E}. Then
|A(q)d | = |{m ⌠X/d | dm = aq2 + 1}|= |{a ⌠X 1/2+Δ | aq2 ⥠â1 (mod d)}|
=
{X 1/2+Δ
d+ O(1) if gcd(d , q2) = 1,
0 else,
=1
Ï(q2)
q2âk=1
(k,q2)=1
|{a ⌠X 1/2+Δ | aq2 ⥠âk (mod d)}|+ O(1)
=|B(q)d |Ï(q2)
+ O(1).
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Type I Sums Continue
Since
|A(q)d | =|B(q)d |Ï(q2)
+ O(1),
we haveâmnâA(q)mâŒM
am =âmâŒM
am|A(q)m| =âmâŒM
am
(|B(q)m|Ï(q2)
+ O(1)
)
=1
Ï(q2)
âmnâB(q)mâŒM
am + O(M)
=1
Ï(q2)
âmnâB(q)mâŒM
am + O
(X 1âΔ/4
Q
)
for M †X 1/2.Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Type II Sums
For type II sums we need deeper arguments.
A large sieve result by Baier and Zhao implies that
âqâPq2âŒQ
âŁâŁâŁâŁâŁâŁâŁâŁâ
mnâA(q)mâŒM
ambn â1
Ï(q2)
âmnâB(q)mâŒM
ambn
âŁâŁâŁâŁâŁâŁâŁâŁ = O
(X 1âΔ/2
Q1/2
)
for M â [X 3/8+2Δ,X 5/8â2Δ].
Thus for most prime squares q2 ⌠Q (all but O(Q1/2XâΔ/4))âŁâŁâŁâŁâŁâŁâŁâŁâ
mnâA(q)mâŒM
ambn â1
Ï(q2)
âmnâB(q)mâŒM
ambn
âŁâŁâŁâŁâŁâŁâŁâŁ = O
(X 1âΔ/4
Q
).
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Arithmetic information
By previous slides, we have type I information
âmnâA(q)mâŒM
am =1
Ï(q2)
âmnâB(q)mâŒM
am + O
(X 1âΔ/4
Q
)
for M †X 1/2 and type II information
âmnâA(q)mâŒM
ambn =1
Ï(q2)
âmnâB(q)mâŒM
ambn + O
(X 1âΔ/4
Q
).
for most q with M â [X 3/8+2Δ,X 5/8â2Δ].
Next task is to splitâ
pâA(q) 1 into type I and type II sums forwhich we use Harman's sieve method.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Sieve Notation
We write
S(E , z) = |{m â E | p | m =â p > z}|.
Then A(q) â© P = S(A(q), 3X 1/2).
Buchstab's identity states that for z > w â„ 1,
S(E , z) = S(E ,w)ââ
wâ€p<z
S(Ep, p).
This holds since by deïżœnitions
S(Ep, p) = |{pn â E | p0 | n =â p0 > p}|= |{n â E | smallest prime factor of n equals p}|.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Decomposing A(q) â© P
So we want information about A(q) â© P = S(A(q), 3X 1/2).
We use Buchstab's identity with w = X 1/4â4Δ in order todecompose
S(A(q), 3X 1/2) = S(A(q),w)ââ
w<p<3X 1/2
S(A(q)p, p)
and further
S(A(q),w) =â
nâA(q)
1ââp1<w
âp1nâA(q)
1 +â
p1<p2<w
âp1p2nâA(q)
1â . . .
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Finding type I and type II sums
So, we ended up with sums of the formâp1<···<pj<w
âp1...pjnâA(q)
1.
If p1 . . . pj †X 1/2, we have a type I sums with
am =
{1 if m = p1 · · · pj with p1 < · · · < pj < w ,
0 otherwise.
If p1 . . . pj > X 1/2, then for some k †j , the productp1 . . . pk â [X 3/8+2Δ,X 5/8â2Δ] since pi < w = X 1/4â4Δ. Thuswe have a type II sum.
Hence
S(A(q),w) = type I sums + type II sums.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Further decomposing
Since
S(A(q), 3X 1/2) = S(A(q),w)ââ
w<p<3X 1/2
S(A(q)p, p),
we still need to handle âw<p<3X 1/2
S(A(q)p, p)
If p > X 3/8+2Δ, we have type II sum. Also if summand isreplaced by S(A(q)p,w) we have type I and II sums as before.
Hence we can apply Buchstab's identity and ïżœnd that
S(A(q), 3X 1/2) = type I/II sums +â
w<p2<p1<X 3/8+2Δ
S(A(q)p1p2 , p2).
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Final steps of the proof
Now we have
S(A(q), 3X 1/2) â„ S(A(q), 3X 1/2)ââ
w<p2<p1<X 3/8+2Δ
S(A(q)p1p2 , p2)
= type I sums + type II sums
=1
Ï(q2)
S(B(q), 3X 1/2)ââ
w<p2<p1<X 3/8+2Δ
S(B(q)p1p2 , p2)
+ O
(X 1âΔ/4
Q
)â„ X
2Q logX> 0
for most q: Using the prime number theorem it is easy to see that
S(B(q), 3X 1/2)ââ
w<p2<p1<X 3/8+2Δ
S(B(q)p1p2 , p2) â„2X
3 logX.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Further thoughts
We have actually proved that for most prime squaresq2 ⌠X 1/2âΔ, there are at least X/(Ï(q2)2 logX ) primes withp ⌠X and p ⥠1 (mod q2).
The residue class 1 is not special here, and the result could beshown even uniformly for any k such that gcd(k, q2) = 1.
Getting over barrier 1/2 needs new ideas - type II informationdisappears at this point.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Summary
Diïżœerent approximations to p = n2 + 1 showed well howinnocent-looking number theoretic problems lead to very deepmethods.
Actually it does not seem likely that any of the approaches canïżœnally settle the conjecture.
Our approach was to show that there are primes p such thatp â 1 possesses a large square factor.
Kaisa MatomÀki Primes of the form aq2 + 1
Approaches to primes of the form n2 + 1Primes of the form aq2 + 1
Thank you
Any questions?
Kaisa MatomÀki Primes of the form aq2 + 1