Sums of Seven Cubes. Samir Siksek. 15 October University of Warwick

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Transcription:

Sums of Seven Cubes Samir Siksek University of Warwick 15 October 2015

Waring, Jacobi and Dase Let k 2. A k-th power means the k-th power of a non-negative integer. Conjecture (Waring, 1770) Every integer is the sum of four squares, nine cubes, 19 biquadrates,... At the request of Jacobi, the famous computer Dase constructed a table showing the least number of positive cubes whose sum is any p < 12000. Dickson

Jacobi s Conjecture on Sums of Cubes Conjecture (Jacobi, 1851) 1 (Waring) Every positive integer is the sum of nine cubes. 2 Every positive integer other than 23 and 239 is the sum of eight cubes. 3 Every positive integer other than is the sum of seven cubes. 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 4 Every positive integer other than 7, 14, 15,..., 5818, 8042 (138 exceptions) is the sum of six cubes. 5 Sufficiently large integers are sums of five cubes.

Conjectures for sums of five/four cubes Conjecture (Romani, 1982) Every positive integer other than 6, 7, 13, 14, 15,..., 1 290 740 (4060 exceptions) is the sum of five cubes. Conjecture (Deshouillers, Hennecart and Landreau, 2000) There are exactly 113 936 676 positive integers which are not the sum of four cubes, the largest of which is 7 373 170 279 850.

Theorems: Nine Cubes Theorem (Wieferich (1908), Kempner (1912)) Every positive integer is the sum of nine cubes. Theoretical part of proof: every positive N > 2.25 10 9 is the sum of nine cubes. Computational part of proof: apply greedy algorithm to show that every N 2.25 10 9 is the sum of nine cubes. Fact: N = 3 N 3 + O(N 2/3 ). Greedy Algorithm: To show that all positive integers K are sums of n cubes, show that all integers C K 2/3 are sums of n 1 cubes. Two iterations of greedy algorithm bring the bound into the range 40000 of von Sterneck s table (1902).

Theorems: Eight Cubes Theorem (Landau (1911), Baer (1913), Dickson (1939)) Every positive integer except 23 and 239 is the sum of eight cubes. Theoretical part of proof: every positive N > 2.26 10 15 is the sum of nine cubes. Computational part of proof: apply greedy algorithm to show that every N 2.26 10 15 is the sum of nine cubes. Miss Evelyn Garbe: Extended von Sterneck s tables to 123, 000.

Theorems: Seven Cubes Theorem (Linnik (1943), Watson (1951)) Every sufficiently large integer is the sum of seven cubes. Theorem (Ramaré (2007)) Every integer > exp(524) ( 3.72 10 227 ) is the sum of seven cubes. Theorem (Deshouillers, Henncart and Landreau, 2000) Every 454 < N exp(78.7) ( 1.51 10 34 ) is the sum of seven cubes. Theorem (Boklan and Elkies (2009), Elkies (2010)) Every positive even integer > 454 is the sum of seven cubes.

Maillet s Identity (1895) Sacrifice two cubes to get a square: (r + x) 3 + (r x) 3 = 2r 3 + 6rx 2. Sacrifice six cubes to get a sum of three squares: (r+x) 3 +(r x) 3 +(r+y) 3 +(r y) 3 +(r+z) 3 +(r z) 3 = 6r 3 +6r(x 2 + y 2 + z 2 }{{} ) More friendly variant: (r + 1 + x) 3 + (r x) 3 + (r + 1 + y) 3 + (r y) 3 + (r + 1 + z) 3 + (r z) 3 = (6r + 3)(r 2 + r + 1 + x 2 + x + y 2 + y + z 2 + z }{{} ). Theorem (Gauss) Let k 0 be even. There exist x, y, z such that x 2 + x + y 2 + y + z 2 + z = k.

(r + 1 + x) 3 + (r x) 3 + (r + 1 + y) 3 + (r y) 3 + (r + 1 + z) 3 + (r z) 3 = (6r + 3)(r 2 + r + 1 + x 2 + x + y 2 + y + z 2 + z). Let N be an odd integer. To show that N is the sum of seven cubes, find t, r, x, y, z such that N 8t 3 = (6r }{{ + 3 } )(r 2 + r + 1 + x 2 + x + y 2 + y + z 2 + z }{{} ). m Choose m such that 1 m 3 (mod 6) is a positive integer (and let r = (m 3)/6); 2 m squarefree; 3 every prime divisor of m/3 is 5 (mod 6). Let t satisfy N 8t 3 (mod m). Then is an even integer. N 8t 3 m r 2 r 1

Well-known lemma Lemma Let 9K/10 N K be an odd integer. Let m satisfy (i) m is squarefree; 3 m; (ii) every prime divisor of m/3 is 5 (mod 6); (iii) 2.63K 1/3 m 2.92K 1/3. Let 0 t < m satisfy t 3 N/8 (mod m). Suppose 0 t < m/10. Then N is the sum of seven cubes. Fundamental Problem with lemma: t can be any integer in [0, m). Standard solution: Perturb the identity to improve the inequalities.

Bombieri s (u 2 v 2 s + wx) 3 + (u 2 v 2 s wx) 3 identity + (v 2 w 2 s + uy) 3 + (v 2 w 2 s uy) 3 + (w 2 u 2 s + vz) 3 + (w 2 u 2 s vz) 3 = 2(u 6 v 6 + v 6 w 6 + w 6 u 2 ) + 6su 2 v 2 w 2 (x 2 + y 2 + z 2 ) Ramaré. To express N as the sum of seven cubes: Let u, v, w 5 (mod 6) be prime, gcd(uvw, N) = 1; Let l satisfy l 3 4N/u 6 v 6 (mod w 2 ), l 3 4N/v 6 w 6 (mod u 2 ), l 3 4N/w 6 u 6 (mod v 2 ). Let s 5 (mod 6) be divisible only by primes 5 (mod 6) such that s l 2 (mod u 2 v 2 w 2 ). Impose several inequalities... A large sieve extension of the Brun-Titchmarsh inequality shows that the existence of suitable u, v, w, s for N > exp(524). Boklan and Elkies. Use variants where x 2 + y 2 + z 2 is replaced by a positive definite ternary form unique in its genus.

Back to fundamentally useless lemma! Lemma Let 9K/10 N K be an odd integer. Let m satisfy (i) m is squarefree; 3 m; (ii) every prime divisor of m/3 is 5 (mod 6); (iii) 2.63K 1/3 m 2.92K 1/3. Let 0 t < m satisfy t 3 N/8 (mod m). Suppose 0 t < m/10. Then N is the sum of seven cubes. Example/Heuristic. Let K = exp(524). Can find 9993 values of m satisfying (i) (iii) with prime divisors 1097. Given N, probability that a given m fails to show that N is the sum of seven cubes is 9/10. The expected number of N for which all these m fail is (K/20) (9/10) 9993 1.03 10 231.

Reformulation with many values of m For x R and m Z 1. Define quotient and remainder by Q(x, m) = x/m, R(x, m) = x Q(x, m) m (R(x, m) [0, m)). Let Bad(m) = {x R : R(x, m) [m/10, m)}.

Reformulation with many values of m For x R and m Z 1. Define quotient and remainder by Q(x, m) = x/m, R(x, m) = x Q(x, m) m (R(x, m) [0, m)). Let Bad(m) = {x R : R(x, m) [m/10, m)}. Lemma Let 9K/10 N K be an odd integer. Let m satisfy (i) (iii). Let t 3 N/8 (mod m). If N is not the sum of seven cubes then t Bad(m).

Reformulation with many values of m For x R and m Z 1. Define quotient and remainder by Q(x, m) = x/m, R(x, m) = x Q(x, m) m (R(x, m) [0, m)). Let Bad(m) = {x R : R(x, m) [m/10, m)}. Lemma Let 9K/10 N K be an odd integer. Let m satisfy (i) (iii). Let t 3 N/8 (mod m). If N is not the sum of seven cubes then t Bad(m). Lemma Let 9K/10 N K be odd. Let W be a set of integers m satisfying (i) (iii). Let M = lcm(w), Bad(W) = m W Bad(m). Let T 3 N/8 (mod M). If N is not the sum of seven cubes then T Bad(W).

Definitions. Let W be a set of m satisfying (i) (iii). Bad(m) = {x R : R(x, m) [m/10, m)} = k Z km + [m/10, m), and M = lcm(w), Bad(W) = m W Bad(m). Heuristic. Given x [0, M), the probability that x Bad(m) is 9/10. The probability that x Bad(W) is (9/10) #W (assuming independence). Expect l([0, M) Bad(W)) M (9/10) #W. Example. Let K = exp(524) 3.72 10 227. Can find 9993 values of m satisfying (i) (iii) with prime divisors 1097. Then M 9.44 10 235. Expect: l([0, M) Bad(W)) 5.25 10 222.

Computing Bad(W) (or [0, M) Bad(W)) Lemma Let [A, B) [0, M). (i) If A / Bad(W) then there is an explicit A < A B such that [A, B) Bad(W) = [A, B) Bad(W). (ii) If A Bad(W) then there is an explicit A < A B such that [A, B) Bad(W) = [A, A ) ( [A, B) Bad(W) ). Heuristic. Number of iterations needed to compute [A, B) Bad(W) is 7(B A)/K 1/3. Example continued. Number of iterations needed to compute [0, M) Bad(W) is 10 161.

The Tower. Let M 0, M 1,..., M r satisfy M i M i+1 and M r = M. Let W i = {m W : m M i }, U i = W i+1 \ W i, p i = M i+1 /M i. π : [0, M i+1 ) [0, M i ), x R(x, M i ). Observe Then π 1 (I ) = p i 1 (km i + I ), I [0, M i ). k=0 [0, M i+1 ) Bad(W i+1 ) = π 1 ( [0, M i ) Bad(W i ) ) Bad(U i ). Heuristic. Number of steps needed to compute Bad(W) is 7 K 1/3 (M 0 + M 1 0.9 #W 0 + M 2 0.9 #W 1 + + M r 0.9 #W r 1 ). Choose M 0,..., M r such that M 0 /K 1/3 is not too big, and the terms M i 0.9 #W i 1 dying quickly.

Example. Let K = exp(524) 3.72 10 227. Can find 9993 values of m satisfying (i) (iii) with prime divisors 1097. Then M 9.44 10 235. Computation shows that [0, M) Bad(W) is the union of 729 intervals and l([0, M) Bad(W)) = 1245835103891227282270190365612369211785920530251524580853973827407412033198232 9 10 1.25 10 78. All intervals are concentrated in small neighbourhoods of (a/q) M with q 42.

Proof Conclusion Let K = exp(524) 3.72 10 227. Recall M 9.44 10 235. Suppose 9K/10 N K is odd and not the sum of seven cubes. Then N 8T 3 (mod M), T Bad(W). So, there is a/q with q 42 such that T = (a/q) M + α, α is small. Let k = qα = qt am Z. Then k qt (mod M) and q 3 N 8(qT ) 3 8k 3 (mod M). But q 3 N 8k 3 < M Hence q 3 N = 8k 3. Summary: If N is not the sum of seven cubes then N is a cube!!!

Theorem Building on Maillet, Linnik, Watson, Cook, McCurley, Bombieri, Ramaré, Boklan, Elkies,... Theorem Every positive integer other than is the sum of seven cubes. 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 Proof took about 18000 hours of computation distributed over 59 processors.

Two Big Questions Can we trust the computation? Why is Bad(W) concentrated around (a/q) M with q small?

An interval in Bad(W) Recall. W be a set of positive integers m satisfying (i) m is squarefree; 3 m; (ii) every prime divisor of m/3 is 5 (mod 6); (iii) 2.63K 1/3 m 2.92K 1/3. and M = lcm(w), Bad(m) = {x R : R(x, m) [m/10, m)}, Bad(W) = m W Bad(m). Observe R(M 1, m) = m 1 [m/10, m). Observe M 1 Bad(W). In fact [M m + m/10, M ) Bad(m). [ ) max(m m + m/10), M Bad(W). m W

Constructing another interval in Bad(W) Observe gcd(m/m, 42) = 1. Thus M/m 6, 7, 8, 9, 10 (mod 42). ( ) M R m, 42 / (5, 11). ( ) M R 42, m = 142 R(M, 42m) = m42 ( ) ( ) Mm 5 R, 42 11 / m, 42 42 m. For small α, Let Then α R ( ) ( ) M 5 42 α, m 11 / m α, 42 42 m α. ( ) 5 17 m, 42 105 m (17/105 = 11/42 1/10). ( ) M ( R 42 α, m / 0, m ). 10 ( M 42 17 105 m, M 42 5 ) 42 m Bad(m).

Constructing another interval in Bad(W) Recall ( M 42 17 105 m, M 42 5 ) 42 m Bad(m). ( M 42 17 105 m, M 42 5 ) 42 m Bad(W). m W 2.63K 1/3 m 2.92K 1/3. ( M 42 4471 10500 K 1/3 M, 42 73 ) 210 K 1/3 Bad(W). }{{} length 0.78K 1/3

Computational Check! One of the 729 intervals that make up Bad(W) is [u, v) where the end points u, v are u =224729153673406083350537419070585735179742636808176263786220 802719358138316832127260626275255886484691752910841237970125 176209271211158960295451324225439375152333282739992912680110 4219646842574053484848749120905514200286748213450123201 1 2, v =224729153673406083350537419070585735179742636808176263786220 802719358138316832127260626275255886484691752910841237970125 176209271211158960295451324225439375152389586874519706498949 5598965866892113341994396851643848724042458952849234990, It turns out that ( M 42 4471 10500 K 1/3, M 42 73 ) 210 K 1/3 [u, v). (4471/10500 73/210) K 1/3 0.99858 v u

Conclusion Thank You!

Ripples Proposition Let a/q (0, 1] be a rational in simplest form with q 11 and 1 a q 1. Let (q 10)/10 < d < q 1 be an integer, and let s be a non-negative integer satisfying Suppose s < q d 1, s < 263 (10d + 10 q). 290 s + 1, s + 2,..., s + d / { } am/m : m W Z/qZ. Let π = 292 100 s q, Π = 263 ( (s + d + 1) 100 q ( a Then π < Π and q M Π K 1/3, a ) q M π K 1/3 1 10 ). Bad(W).

Gaps Proposition Suppose the longest sequence s + 1, s + 2,..., s + d / satisfies d < (q 10)/10. Let µ = 38398 725 { } am/m : m W Z/qZ. ( ) 1 (d + 1). 20 2q Then ( a q M µk 1/3, ) a 1/3 M + µk q Bad(W) =. Heuristic. If #W 10 and q 50 then Bad(W) is empty in a neighbourhood of am/q.

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