Lucas Congruences. A(n) = Pure Mathematics Seminar University of South Alabama. Armin Straub Oct 16, 2015 University of South Alabama.

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1 Pure Mathematics Seminar University of South Alabama Oct 6, 205 University of South Alabama A(n) = n k=0 ( ) n 2 ( n + k k k ) 2, 5, 73, 445, 3300, 89005, ,... Arian Daneshvar Amita Malik Zhefan Wang Pujan Dave (Illinois Geometry Lab, UIUC, Fall 204) / 25

2 Illinois Geometry Lab, UIUC, Fall 204 Arian Daneshvar Amita Malik Zhefan Wang Pujan Dave semester-long project to introduce undergraduate students to research graduate student team leader: Amita Malik 2 / 25

3 Rough outline introducing Apéry-like numbers Lucas-type congruences applications primes never dividing Apéry-like numbers periodicity modulo p a little more on supercongruences (time permitting) 3 / 25

4 Positivity of rational functions Let us begin with an open problem: CONJ Kauers- Zeilberger 2008 All Taylor coefficients of the following function are positive: (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. 4 / 25

5 Positivity of rational functions Let us begin with an open problem: CONJ Kauers- Zeilberger 2008 PROP S-Zudilin 205 All Taylor coefficients of the following function are positive: (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. The diagonal coefficients of the Kauers Zeilberger function are D(n) = n ( ) n 2 ( ) 2k 2. k n k=0 D(n) is an example of an Apéry-like sequence. 4 / 25

6 Positivity of rational functions Let us begin with an open problem: CONJ Kauers- Zeilberger 2008 PROP S-Zudilin 205 All Taylor coefficients of the following function are positive: (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. The diagonal coefficients of the Kauers Zeilberger function are D(n) = n ( ) n 2 ( ) 2k 2. k n k=0 D(n) is an example of an Apéry-like sequence. Q S-Zudilin 205 Can we conclude the conjectured positivity from the positivity of D(n) together with the (obvious) positivity of (x+y+z)+2xyz? 4 / 25

7 The Riemann zeta function The Riemann zeta function is the analytic continuation of ζ(s) = n= n s = p prime p s. Its zeros and values are fundamental, yet mysterious to this day. CONJ RH If ζ(s) = 0 then s { 2, 4,...} or Re (s) = 2. 5 / 25

8 The Riemann zeta function The Riemann zeta function is the analytic continuation of ζ(s) = n= n s = p prime p s. Its zeros and values are fundamental, yet mysterious to this day. CONJ RH If ζ(s) = 0 then s { 2, 4,...} or Re (s) = 2. THM Euler 734 ζ(2) = π2 π4, ζ(4) = 6 90,..., ζ(2n) = ( )n+ (2π) 2n B 2n 2(2n)! CONJ The values ζ(3), ζ(5), ζ(7),... are all transcendental. 5 / 25

9 The Riemann zeta function The Riemann zeta function is the analytic continuation of ζ(s) = n= n s = p prime p s. Its zeros and values are fundamental, yet mysterious to this day. CONJ RH If ζ(s) = 0 then s { 2, 4,...} or Re (s) = 2. THM Euler 734 ζ(2) = π2 π4, ζ(4) = 6 90,..., ζ(2n) = ( )n+ (2π) 2n B 2n 2(2n)! CONJ The values ζ(3), ζ(5), ζ(7),... are all transcendental. THM Apéry 78 ζ(3) is irrational. 5 / 25

10 Apéry numbers and the irrationality of ζ(3) The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k satisfy k=0 (n + ) 3 A(n + ) = (2n + )(7n 2 + 7n + 5)A(n) n 3 A(n ). 6 / 25

11 Apéry numbers and the irrationality of ζ(3) The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k satisfy k=0 (n + ) 3 A(n + ) = (2n + )(7n 2 + 7n + 5)A(n) n 3 A(n ). THM Apéry 78 ζ(3) = n= n 3 is irrational. proof The same recurrence is satisfied by the near -integers n ( ) n 2 ( ) n + k 2 n B(n) = k k k j 3 + ( ) m 2m 3( n m k=0 j= m= )( n+m m ). Then, B(n) A(n) ζ(3). But too fast for ζ(3) to be rational. 6 / 25

12 Zagier s search and Apéry-like numbers Recurrence for Apéry numbers is the case (a, b, c) = (7, 5, ) of Q Beukers, Zagier (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n cn 3 u n. Are there other tuples (a, b, c) for which the solution defined by u = 0, u 0 = is integral? 7 / 25

13 Zagier s search and Apéry-like numbers Recurrence for Apéry numbers is the case (a, b, c) = (7, 5, ) of Q Beukers, Zagier (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n cn 3 u n. Are there other tuples (a, b, c) for which the solution defined by u = 0, u 0 = is integral? Essentially, only 4 tuples (a, b, c) found. (Almkvist Zudilin) 4 hypergeometric and 4 Legendrian solutions (with generating functions 3F 2 ( 2, α, α, ) 4C αz, with α = 2, 3, 4, 6 and C α = 2 4, 3 3, 2 6, ) 6 sporadic solutions ( ) α, α 2 C α z 2 F C α z, C α z Similar (and intertwined) story for: (n + ) 2 u n+ = (an 2 + an + b)u n cn 2 u n (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n n(cn 2 + d)u n (Beukers, Zagier) (Cooper) 7 / 25

14 The six sporadic Apéry-like numbers (a, b, c) A(n) ( ) n 2 ( ) n + k 2 (7, 5, ) Apéry numbers k n k ( ) n 2 ( ) 2k 2 (2, 4, 6) k n k ( ) n 2 ( )( ) 2k 2(n k) (0, 4, 64) Domb numbers k k n k k ( )( ) n n + k (3k)! (7, 3, 8) ( ) k 3 n 3k Almkvist Zudilin numbers 3k n k! 3 k ( ) n 3 (( ) ( )) 4n 5k 4n 5k (, 5, 25) ( ) k + k 3n 3n k ( ) n 2 ( )( )( ) n k k + l (9, 3, 27) k l l n k,l 8 / 25

15 Apéry-like numbers and modular forms The Apéry numbers A(n) satisfy, 5, 73, 45,... η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 2 (τ)η 2 ) n (6τ) A(n) (6τ) η 2 (2τ)η 2. (3τ) n 0 modular form + 5q + 3q q 3 + O(q 4 ) modular function q 2q q 3 + O(q 4 ) q = e 2πiτ 9 / 25

16 Apéry-like numbers and modular forms The Apéry numbers A(n) satisfy, 5, 73, 45,... η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 2 (τ)η 2 ) n (6τ) A(n) (6τ) η 2 (2τ)η 2. (3τ) n 0 FACT modular form + 5q + 3q q 3 + O(q 4 ) modular function q 2q q 3 + O(q 4 ) q = e 2πiτ Not at all evidently, such a modular parametrization exists for all known Apéry-like numbers! 9 / 25

17 Apéry-like numbers and modular forms The Apéry numbers A(n) satisfy, 5, 73, 45,... η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 2 (τ)η 2 ) n (6τ) A(n) (6τ) η 2 (2τ)η 2. (3τ) n 0 FACT modular form + 5q + 3q q 3 + O(q 4 ) modular function q 2q q 3 + O(q 4 ) q = e 2πiτ Not at all evidently, such a modular parametrization exists for all known Apéry-like numbers! As a consequence, with z = 34x + x 2, Context: A(n)x n = n 0 f(τ) x(τ) y(x) 7 x z 4 2( + x + z) 3/2 3 F 2 ( 2, 2, 2, 024x ( x + z) 4 modular form of (integral) weight k modular function such that y(x(τ)) = f(τ) Then y(x) satisfies a linear differential equation of order k +. ). 9 / 25

18 Lucas congruences The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k k=0 satisfy the Lucas congruences (Gessel 982) A(n) A(n 0 )A(n ) A(n r ) (mod p), where n i are the p-adic digits of n. 0 / 25

19 Lucas congruences The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k k=0 satisfy the Lucas congruences (Gessel 982) A(n) A(n 0 )A(n ) A(n r ) (mod p), where n i are the p-adic digits of n. Lucas showed the beautiful congruences ( ) ( )( ) n n0 n k k 0 k ( nr k r ) (mod p), where n i, respectively k i, are the p-adic digits of n and k. 0 / 25

20 Lucas congruences The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k k=0 satisfy the Lucas congruences (Gessel 982) A(n) A(n 0 )A(n ) A(n r ) (mod p), where n i are the p-adic digits of n. Lucas showed the beautiful congruences ( ) ( )( ) n n0 n k k 0 k ( nr k r ) (mod p), where n i, respectively k i, are the p-adic digits of n and k. THM Malik S 205 Every sporadic sequence satisfies these Lucas congruences modulo every prime. 0 / 25

21 Primes not dividing Apéry numbers CONJ Rowland Yassawi There are infinitely many primes p such that p does not divide any Apéry number A(n). Such as p = 2, 3, 7, 3, 23, 29, 43, 47,... Recall that the Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k k=0 satisfy the Lucas congruences A(n) A(n 0 )A(n ) A(n r ) (mod p). / 25

22 Primes not dividing Apéry numbers CONJ Rowland Yassawi There are infinitely many primes p such that p does not divide any Apéry number A(n). Such as p = 2, 3, 7, 3, 23, 29, 43, 47,... EG The values of Apéry numbers A(0), A(),..., A(6) modulo 7 are, 5, 3, 3, 3, 5,. Recall that the Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k k=0 satisfy the Lucas congruences A(n) A(n 0 )A(n ) A(n r ) (mod p). / 25

23 Primes not dividing Apéry numbers CONJ Rowland Yassawi There are infinitely many primes p such that p does not divide any Apéry number A(n). Such as p = 2, 3, 7, 3, 23, 29, 43, 47,... EG The values of Apéry numbers A(0), A(),..., A(6) modulo 7 are, 5, 3, 3, 3, 5,. Hence, the Lucas congruences imply that 7 does not divide any Apéry number. Recall that the Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k k=0 satisfy the Lucas congruences A(n) A(n 0 )A(n ) A(n r ) (mod p). / 25

24 Primes not dividing Apéry numbers, cont d CONJ DDMSW 205 The proportion of primes not dividing any Apéry number A(n) is e / %. 2 / 25

25 Primes not dividing Apéry numbers, cont d CONJ DDMSW 205 The proportion of primes not dividing any Apéry number A(n) is e / %. Heuristically, combine Lucas congruences, palindromic behavior of Apéry numbers, that is A(n) A(p n) ( and e /2 = lim (p+)/2. p p) (mod p), 2 / 25

26 Primes not dividing Apéry numbers, cont d CONJ DDMSW 205 The proportion of primes not dividing any Apéry number A(n) is e / %. Heuristically, combine Lucas congruences, palindromic behavior of Apéry numbers, that is A(n) A(p n) ( and e /2 = lim (p+)/2. p p) (mod p), proportion of primes not dividing any Apéry number / 25

27 Primes not dividing Apéry numbers, cont d 2 The primes below 00 not dividing sporadic sequences, as well as the proportion of primes below 0, 000 not dividing any term (δ) 2, 5, 7,, 3, 9, 29, 4, 47, 6, 67, 7, 73, 89, (η) 2, 3, 7, 9, 23, 3, 47, 53, (α) 3, 5, 3, 7, 29, 3, 37, 4, 43, 47, 53, 59, 6, 67, 7, 83, (ɛ) 3, 7, 3, 9, 23, 29, 3, 37, 43, 47, 6, 67, 73, 83, (ζ) 2, 5, 7, 3, 7, 9, 29, 37, 43, 47, 59, 6, 67, 7, 83, (γ) 2, 3, 7, 3, 23, 29, 43, 47, 53, 67, 7, 79, 83, / 25

28 Primes not dividing Apéry numbers, cont d 2 The primes below 00 not dividing sporadic sequences, as well as the proportion of primes below 0, 000 not dividing any term (δ) 2, 5, 7,, 3, 9, 29, 4, 47, 6, 67, 7, 73, 89, (η) 2, 3, 7, 9, 23, 3, 47, 53, (α) 3, 5, 3, 7, 29, 3, 37, 4, 43, 47, 53, 59, 6, 67, 7, 83, (ɛ) 3, 7, 3, 9, 23, 29, 3, 37, 43, 47, 6, 67, 73, 83, (ζ) 2, 5, 7, 3, 7, 9, 29, 37, 43, 47, 59, 6, 67, 7, 83, (γ) 2, 3, 7, 3, 23, 29, 43, 47, 53, 67, 7, 79, 83, / 25

29 Primes not dividing Apéry numbers, cont d 2 The primes below 00 not dividing sporadic sequences, as well as the proportion of primes below 0, 000 not dividing any term (δ) 2, 5, 7,, 3, 9, 29, 4, 47, 6, 67, 7, 73, 89, (η) 2, 3, 7, 9, 23, 3, 47, 53, (α) 3, 5, 3, 7, 29, 3, 37, 4, 43, 47, 53, 59, 6, 67, 7, 83, (ɛ) 3, 7, 3, 9, 23, 29, 3, 37, 43, 47, 6, 67, 73, 83, (ζ) 2, 5, 7, 3, 7, 9, 29, 37, 43, 47, 59, 6, 67, 7, 83, (γ) 2, 3, 7, 3, 23, 29, 43, 47, 53, 67, 7, 79, 83, THM Malik S 205 For any prime p 3, we have that, modulo p, { ( p ) ( ) A η p/5 ( p/3 p/5 )3, if p, 2, 4, 8 (mod 5), 3 0, otherwise. We therefore expect the proportion of primes not dividing any A η (n) to be 2 e / %. 3 / 25

30 Modular (super)congruences THM Ahlgren Ono 00 The Apéry numbers satisfy ( ) p A a(p) (mod p 2 ) 2 with a(n)q n = η 4 (2τ)η 4 (4τ). n= conjectured by Beukers 87, and proved modulo p similar congruences modulo p for other Apéry-like numbers 4 / 25

31 Periodicity of residues THM Gessel 82 A(n) { (mod 8), if n even, 5 (mod 8), if n odd. conjectured by Chowla Cowles Cowles 80 not eventually periodic modulo 6 (Rowland Yassawi 3) 5 / 25

32 Periodicity of residues THM Gessel 82 A(n) { (mod 8), if n even, 5 (mod 8), if n odd. conjectured by Chowla Cowles Cowles 80 not eventually periodic modulo 6 (Rowland Yassawi 3) PROP Gessel 82 If C(n) satisfies Lucas congruences modulo p and is eventually periodic modulo p, then C(n) C() n (mod p) for all n = 0,,..., p. 5 / 25

33 Periodicity of residues THM Gessel 82 A(n) { (mod 8), if n even, 5 (mod 8), if n odd. conjectured by Chowla Cowles Cowles 80 not eventually periodic modulo 6 (Rowland Yassawi 3) PROP Gessel 82 If C(n) satisfies Lucas congruences modulo p and is eventually periodic modulo p, then C(n) C() n (mod p) for all n = 0,,..., p. EG For the Almkvist Zudilin sequence Z(n) = n ( )( ) n n + k (3k)! ( 3) n 3k 3k n k! 3, k=0 Z(3) Z() 3 = 24. So can be periodic modulo p only for p = 2, 3. 5 / 25

34 Periodicity of residues THM DDMSW 205 The Almkvist Zudilin numbers n ( )( ) n n + k (3k)! Z(n) = ( 3) n 3k 3k n k! 3 k=0 satisfy the congruences Z(n) { (mod 8), if n even, 5 (mod 8), if n odd. This can be proved using computer algebra in two steps. 6 / 25

35 Periodicity of residues THM DDMSW 205 The Almkvist Zudilin numbers n ( )( ) n n + k (3k)! Z(n) = ( 3) n 3k 3k n k! 3 k=0 satisfy the congruences Z(n) { (mod 8), if n even, 5 (mod 8), if n odd. This can be proved using computer algebra in two steps. LEM S 204 The Almkvist Zudilin numbers are the diagonal coefficients of (x + x 2 + x 3 + x 4 ) + 27x x 2 x 3 x 4. That is, Z(n) equals the coefficient of (x x 2 x 3 x 4 ) n. 6 / 25

36 Diagonals The diagonal of a multivariate series F (x,..., x d ) = a(n,..., n d )x n xn d d, n,...,n d 0 is the univariate function n 0 a(n,..., n)zn. The diagonal of an algebraic function is D-finite. Gessel Zeilberger Lipshitz 7 / 25

37 Diagonals The diagonal of a multivariate series F (x,..., x d ) = a(n,..., n d )x n xn d d, n,...,n d 0 is the univariate function n 0 a(n,..., n)zn. The diagonal of an algebraic function is D-finite. Gessel Zeilberger Lipshitz EG x y has diagonal coefficients ( 2n n ). 7 / 25

38 Diagonals The diagonal of a multivariate series F (x,..., x d ) = a(n,..., n d )x n xn d d, n,...,n d 0 is the univariate function n 0 a(n,..., n)zn. The diagonal of an algebraic function is D-finite. Gessel Zeilberger Lipshitz EG x y = (x + y) k k=0 has diagonal coefficients ( 2n n ). 7 / 25

39 Diagonals The diagonal of a multivariate series F (x,..., x d ) = a(n,..., n d )x n xn d d, n,...,n d 0 is the univariate function n 0 a(n,..., n)zn. The diagonal of an algebraic function is D-finite. Gessel Zeilberger Lipshitz EG x y = (x + y) k k=0 has diagonal coefficients ( 2n n ). The diagonal is n=0 ( ) 2n z n =. n 4z 7 / 25

40 Finite state automata EG x y = (x + y) n n=0 has diagonal coefficients ( 2n n ). The diagonal is n=0 ( ) 2n z n =. n 4z 8 / 25

41 Finite state automata EG x y = (x + y) n n=0 has diagonal coefficients ( 2n n ). The diagonal is n=0 ( ) 2n z n =. n 4z The diagonal of a rational function F (x, y) is always algebraic. To see this, express the diagonal as 2πi x =ε F (x, z x ) dx x. Not true for more than two variables. 8 / 25

42 Finite state automata EG x y = (x + y) n n=0 has diagonal coefficients ( 2n n ). The diagonal is n=0 ( ) 2n z n =. n 4z The diagonal of a rational function F (x, y) is always algebraic. To see this, express the diagonal as 2πi x =ε F (x, z x ) dx x. Not true for more than two variables. However: (Furstenberg 67, Deligne 84 and Denef Lipshitz 87) THM Diagonals of algebraic functions in Z p [[x,..., x d ]] are algebraic over Z p (z). Equivalently, the diagonal coefficients modulo p r are generated by a finite state automaton. 8 / 25

43 Finite state automata Recall: the AZ numbers ( ) n Z(n) are the diagonal coefficients of (x + x 2 + x 3 + x 4 ) + 27x x 2 x 3 x 4. 9 / 25

44 Finite state automata Recall: the AZ numbers ( ) n Z(n) are the diagonal coefficients of (x + x 2 + x 3 + x 4 ) + 27x x 2 x 3 x 4. Rowland Yassawi (203) give an algorithm to compute a finite state automaton for these numbers modulo 7 (or any p r ). 0,4,5,6 4 2 For instance: Z(63) = Z(20 base 7 ) (mod 7). 2,3,3 Of course, modulo primes it is easier to use Lucas congruences.,3 2 0,4,5,6 2 0,4,5,6 9 / 25

45 Finite state automata Null The Almkvist Zudilin numbers Z(n) modulo 8: This automatically generated automaton can, of course, be simplified / 25

46 Finite state automata Null The Almkvist Zudilin numbers Z(n) modulo 8: This automatically generated automaton can, of course, be simplified The automaton makes it obvious that, indeed, { (mod 8), if n even, Z(n) 5 (mod 8), if n odd. 20 / 25

47 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). 2 / 25

48 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). 2 / 25

49 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). 2 / 25

50 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). EG For primes p, simple combinatorics proves the congruence ( ) 2p = ( )( ) p p + (mod p 2 ). p k p k k For p 5, Wolstenholme s congruence shows that, in fact, ( ) 2p 2 (mod p 3 ). p 2 / 25

51 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). The congruences a(mp r ) a(mp r ) modulo p r occur frequently: a(n) = tr A n with A Z d d Arnold 03, Zarelua 04,... 2 / 25

52 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). The congruences a(mp r ) a(mp r ) modulo p r occur frequently: a(n) = tr A n with A Z d d Arnold 03, Zarelua 04,... realizable sequences a(n), i.e., for some map T : X X, a(n) = #{x X : T n x = x} points of period n Everest van der Poorten Puri Ward 02, Arias de Reyna 05 2 / 25

53 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). EG Mathematica 7 miscomputes A(n) = n ( ) 2 ( ) 2 n n + k for n > k k k=0 A(5 3 ) = about 2000 digits...about 8000 digits Weirdly, with this wrong value, one still has A(5 3 ) A(5 2 ) (mod 6 ). 2 / 25

54 Supercongruences for Apéry-like numbers Conjecturally, supercongruences like A(mp r ) A(mp r ) (mod p 3r ) Robert Osburn (University of Dublin) Brundaban Sahu (NISER, India) hold for all Apéry-like numbers. Osburn Sahu 09 Current state of affairs for the six sporadic sequences from earlier: (a, b, c) (7, 5, ) (2, 4, 6) (0, 4, 64) (7, 3, 8) (, 5, 25) (9, 3, 27) A(n) ( n 2 ( n+k ) 2 k) Beukers, Coster k ( n ) 2 ( 2k k n k ( n k k n ) 2 Osburn Sahu S 4 ) 2 ( 2k )( 2(n k) ) k k ( )k 3 n 3k( n k ( )k( ) ( n 3 (4n 5k k k,l ( n k) 2 ( n l)( k l n k Osburn Sahu )( n+k ) (3k)! 3k n k! 3 open 3n )( k+l ) n ) + ( 4n 5k 3n modulo p 3 ) ) Osburn Sahu S 4 open Amdeberhan Tauraso 5 22 / 25

55 Apéry numbers as diagonals CONJ Christol 990 Every holonomic integer sequence with at most exponential growth is the diagonal of a rational function. 23 / 25

56 Apéry numbers as diagonals CONJ Christol 990 Every holonomic integer sequence with at most exponential growth is the diagonal of a rational function. THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x / 25

57 Apéry numbers as diagonals CONJ Christol 990 Every holonomic integer sequence with at most exponential growth is the diagonal of a rational function. THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4. Univariate generating function: A(n)x n = n 0 where z = 34x + x 2. 7 x z 4 2( + x + z) 3/2 3 F 2 ( 2, 2, 2, 024x ( x + z) 4 ), 23 / 25

58 Apéry numbers as diagonals CONJ Christol 990 Every holonomic integer sequence with at most exponential growth is the diagonal of a rational function. THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4. Univariate generating function: A(n)x n = n 0 7 x z 4 2( + x + z) 3/2 3 F 2 ( 2, 2, 2, 024x ( x + z) 4 ), where z = 34x + x 2. Well-developed theory of multivariate asymptotics e.g., Pemantle Wilson 23 / 25

59 Apéry numbers as diagonals CONJ Christol 990 Every holonomic integer sequence with at most exponential growth is the diagonal of a rational function. THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4. Let A(n) be the coefficient of x n = x n xn 4 4. Then, for p 5, we have the multivariate supercongruences A(np r ) A(np r ) (mod p 3r ). 23 / 25

60 Apéry numbers as diagonals CONJ Christol 990 Every holonomic integer sequence with at most exponential growth is the diagonal of a rational function. THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4. Let A(n) be the coefficient of x n = x n xn 4 4. Then, for p 5, we have the multivariate supercongruences A(np r ) A(np r ) (mod p 3r ). Numerical evidence suggests the same congruences for (x + x 2 + x 3 + x 4 ) + 27x x 2 x 3 x / 25

61 Some of many open problems Supercongruences for all Apéry-like numbers proof of all the classical ones uniform explanation, proofs not relying on binomial sums Apéry-like numbers as diagonals find minimal rational functions extend supercongruences any structure? polynomial analogs of Apéry-like numbers find q-analogs (e.g., for Almkvist Zudilin sequence) q-supercongruences is there a geometric picture? Many further questions remain. is the known list complete? Apéry-like numbers as diagonals and multivariate supercongruences higher-order analogs, Calabi Yau DEs modular supercongruences Beukers 87, Ahlgren Ono 00 ( ) p A a(p) (mod p 2 ), a(n)q n = η 4 (2τ)η 4 (4τ) 2... n= 24 / 25

62 THANK YOU! Slides for this talk will be available from my website: A. Malik, A. Straub Divisibility properties of sporadic Apéry-like numbers Preprint, 205 A. Straub Multivariate Apéry numbers and supercongruences of rational functions Algebra & Number Theory, Vol. 8, Nr. 8, 204, p R. Osburn, B. Sahu, A. Straub Supercongruences for sporadic sequences to appear in Proceedings of the Edinburgh Mathematical Society, 204 A. Straub, W. Zudilin Positivity of rational functions and their diagonals Journal of Approximation Theory (special issue dedicated to Richard Askey), Vol. 95, 205, p / 25