Bernoulli Numbers*

Computation of Tangent, Euler, and Bernoulli Numbers*

By Donald E. Knuth and Thomas J. Buckholtz

Abstract. Some elementary methods are described which may be used to calculate tangent numbers, Euler numbers, and Bernoulli numbers much more easily and rapidly on electronic computers than the traditional recurrence relations which have been used for over a century. These methods have been used to prepare an accompanying table which extends the existing tables of these numbers. Some the- orems about the periodicity of the tangent numbers, which were suggested by the tables, are also proved.

1. Introduction. The tangent numbers Tn, Euler numbers En, and Bernoulli numbers Bn, are defined to be the coefficients in the following power series:

(1) tan z = To/0! + T&/V. + T2z2/2\ + • • • = Z Tnzn/n\ ,

(2) sec z = E0/0\ + Eiz/V. + E2z2/2\ +•••«= 5Z Enzn/n\ ,

(3) z/(ez - 1) = tfo/O! + Biz/1\ + ^2272! + ••• = £ Bnzn/n\ .

Much of the older mathematical literature uses a slightly different notation for these numbers, to take account of the zero coefficients. Thus we find many papers where tan z is written Tiz + T2z3/3\ + T-¡zb/5\ + - - -, sec z is written Eo + Eiz2/2\ + E2z4/Al + ■ ■-, and z/ (e* - 1) is written 1 - z/2 + Biz2/2l - B^/Al + Bsz6/6l ••'. Some other authors have used essentially the notation defined above but with different signs; in particular our E2n is often accompanied by the sign ( — 1)". In Section 2 we present simple methods for computing T„, En, and Bn which are readily adapted to electronic computers, and in Section 3 more details of the computer program are explained. A table of Tn and En for n ^ 120, and Bn for n ^ 250, is appended to this paper, thereby extending the hitherto published values of Tn for n á 60 [6], En for n á 100 [2, 3], and Bn for n ^ 220 [7, 4]. Using the methods of this paper it is not difficult to extend the tables much further, and the authors have submitted a copy of the values of Tn On gj 835), En (n ^ 808), Bn (n ^ 836) to the Unpublished Mathematical Tables repository of this journal. Section 4 shows how the formulas of Section 2 lead to some simple proofs of arithmetical properties of these numbers.

2. Formulas for Computation. The traditional method of calculating Tn and En is to use recurrence relations, such as the following: Let cos z = 2~2n^o Cnzn/n;

Received February 6, 1967. * Supported in part by NSF Grant GP 3909. 663

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then the coefficient of zn/n\ in (tan z) (cos z) is

T,(nk)TkCn-k

and in (sec z) (cos z) it is

Z{nk)EkCn-k.

Hence, making use of the fact that T2n = E2n+i = 0, we have the recurrence relations (4)(2» + >>, - (2»3+l)r. + ■■■+ (-irfc + ;)r«, _ 1, (« (^-(ïH+'-' +'-'K^K-0- ">o- The disadvantage of these formulas is that the binomial coefficients as well as the numbers Tn, En become very large when n is large, so a time-consuming multiplica- tion of multiple-precision numbers is implied. As Lehmer [4] has observed, we may simplify the calculations if we remember the values of (2\+1K (2;>' so that when n increases by 1 we need only multiply Í2n + 1 \ k n by (2n + 2) (2n + 3) (2n + 2 - k) (2n + 3 - k) to get the next value ; but the method to be described here is even simpler and has other advantages. The tangent numbers may be evaluated by noting that D(t&nnz) is n tan71-1 z (1 + tan2 z) ; hence the nth derivative of tan z is a polynomial in tan z. We have Z>n(tan z) = P„(tan z), where the polynomials Pn(x) are defined by

(6) Pi(x) = X , Pn+lix) = (1 + X2)Pn'ix) . Thus if we write

7>"(tan z) = Tno + Tni tan z + Tn2 tan2 z + ■■ - the coefficients Tnk satisfy the recurrence equation

(7) Tok = oik ; Tn+i,k = (k — l)Tn,k-i + (k + l)Tn,k+i ■ Since Tn = Z)"(tan z)\z=o = Tno, and since Tnk is zero except for at most On + 3)/2 values of k, formula (7) shows that the calculation of all T„+i,k from the values of Tn,k essentially requires only On + 2)/2 multiplications of a small number k by a

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Tn+2,k = (k - 2)(k - l)Tn,k-2 + 2k2Tn,k + (k + l)(k + 2)Tn.k+2 but a count of the operations involved shows this provides little if any improve- ment over (7), and so the simpler form (7) is preferable. Similarly, we have D(sec z tan" z) = sec z (n tan"-1 z + (n + l)tann+1 z), hence if we write

(8) 7)"(sec z) = (sec z) (En0 + Eni tan z + En2 tan2 z + • • • ) we have the recurrence

(9) Eok = Ôok; En+itk = kEn¡k_i + (k + l)En,k+i. Since En = En0, this relation yields an efficient method for calculating the Euler numbers. A somewhat similar recurrence relation was used by Joffe [3] to calculate Euler numbers; his method requires essentially the same amount of computation, but as explained in the next section there is a way to modify (9) to obtain a con- siderable advantage. The identities tan (7r/4 + z/2) = tan z + sec z and Z)n(tan (v/A + z/2)) = 2-nP„(tan (7r/4 + z/2)) imply that the sums of the numbers Tnk have a very simple form:

(10) 2-"P„(l) = 2~nE Tnk= if" n eVf? ' k^o [Tn , wodd . This relation can be used to advantage when both E„ and Tn are being calculated. The definition of tan z implies

sin z (eiz - e~iz) 1 ( 2iz . \ l( 2iz Aiz . \ tan z = -= -:-— = -1 IT--iz I = -1 -T-.--r--iz ) cosz i(e" + e—) 2 V»2 + l / z \e2™ _ 1 eilz - 1 J

= -(-iz+Tl ((2iz)n - (Aiz)n)Bn/n\) ; Z \ n£0 / and by equating coefficients we obtain the well-known identity

(11) Bn = -i~nnTn-i/2n(2n - 1) , n > 1 . Hence, the Bernoulli numbers may be obtained from the tangent numbers by a calculation which (on a binary computer) is especially simple. The celebrated von Staudt-Clausen theorem [8, 1] states that

(12) B2n= C2n- Z I p prime '.(p—1) \2n y where C2n is an integer. The table appended to this paper expresses Bn in this form, and, as shown below, the calculation of (11) may be carried out without any multiple-precision division.

3. Details of the Computation. By the recurrence (7) we may discard the value of Tn,k once T„+i,k+i has been calculated, so only about n of the values T„fk need

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to be retained in the computer memory at any one time. A further technique can be employed when the memory size has been exceeded; for example, suppose we start with the computation of Tnk for n %. A: k = 0 fc = 1 k = 2 fc = 3 k = 4 k = 5 n = 0 0 1 n = 1 1 0 1 n = 2 0 2 0 2 n = 3 2 0 8 0 6 n = A 0 16 0 40 0 24 and suppose that very little memory space is available, so that we cannot com- pletely evaluate all of the entries for n = 5 ; we might obtain n = 5 16 0 136 0 240 0 * where "*" denotes an unknown value. The calculation may still proceed, keeping track of unknown values : n = 6 0 272 0 1232 0 * ii = 7 272 0 3968 0 * n = 8 0 7936 0 * n = 9 7936 0 * etc. In this way we may compute the values of about twice as many tangent numbers as were produced before overflow occurred, avoiding much of the calculation of the Tn,k. Since the numbers Tn become very large (Tss¡, has 1866 digits, and Tn is asymp- totically 2n+2n\/vn+l when n is odd), care needs to be taken for storage allocation of the numbers Tn.k if we are to make efficient use of memory space. The program we prepared makes use of two rather small areas of memory (say A and B) each of which is capable of holding any one of the numbers T„,k, plus a large number of consecutive locations used for all the remaining values. By sweeping cyclically through this large memory area, it is possible to store and retrieve the values in a simple manner. For the sake of illustration let us suppose the word size of our computer is very small, so that only one decimal digit may be stored per word ; and suppose there are just 14 words of memory used for the table of Tn¡k. After the calculation of the values for n = A, the memory might have the following configuration : l~6~i- [116 1, | 4 | 0 | , | 2 | 4 | . | , | 8 | , | (13) î Î P Q Here P and Q represent variables in the program that point to the current places of interest in the memory; P points to the number that will be accessed next, and Q points to the place where the next value is to be written. Only locations from P to Q contain information that will be used subsequently by the program. The sym- bols "." and "," represent special negative codes in the table which delimit the numbers in an obvious fashion. As we begin the calculation for n = 5, we set area A to zero and a variable k to 1. The basic cycle is then :

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(a) Set area B to k times the next value indicated by P, and move P to the right. (b) Store the value of A + B into the locations indicated by Q, and move Q to the right. (c) Transfer the contents of B to area A. (d) Increase fc by 2. In the case of (13) we would change the memory configuration to I 6 | . | 1 | 6 | , | 4 | 0 | , | 2 | 4 | . | 1 | 6 | , | (14) î î Q P k = 3 A = 16 B = 16 Notice that the value 16 has been stored, the pointer Q has moved to the right and (treating the memory as a circular store) then to the far left. The next two itera- tions of steps (a)-(d) give | 1 | 3 | 6 [ , | 2 | 4 | 0 | , | 2 | 4 | . | 1 | 6 | , | (15) T î Q P k = 7 A = 120 I B = 120 Now since the terminating "." was sensed, the program attempts to store the value from area A ; but since this would make pointer Q pass P, the "memory overflow" condition is sensed, and the memory configuration becomes |~1~T3 | 6 | , | 2 | 4 | 0 | , |* | 2 | 0 | 1 | 6 | , | (16) î î Q P where "*" is another internal code symbol. The computation for n = 6 is similar but it uses a different initialization since n is even; after n = 6 has been processed we would have J2[3|2|,|*|4¡0|,|*|2|7|2|,|1| (17) T î Q P and so on. The above discussion has been slightly simplified for purposes of exposition. In the actual program, it is preferable to keep the numbers stored with least significant digit first, so that for example (16) would really be 16|3|1|,|0|4|2|,|*|2|1|6[1|,| (18) î î Q P in order to simplify the multiple-precision operations. A few other changes in the sequence of operations were made in order to use memory a little more efficiently (for example the value Tno need never be retained). A similar method may be used for En. This arrangement of the computation gives a substantial advantage over Joffe's method [3] because of the "*", and it

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also has advantages over (10) for the same reason. It remains to consider the calculation of the Bernoulli number B2n from T2n-i. Consider formula (12) ; if p is an odd prime, 2P-1 = 1 (modulo p), hence if (p — l)\2w, then 22" — 1 is divisible by p. So we first compute the integer (2n)(22n)(22" A (19) n= (-iy x2nT2n-i + Z . p prime \(j>—1) \2n V by referring to an auxiliary table of primes that may be calculated at the beginning of the program. Then it is merely a question of computing (20) C2n = N/22n(22n - 1) = N/2in + N/2«n + N/2Sn + The calculation of N/2k is of course merely a "shift right" operation in a binary computer, so all the terms of the infinite series on the right side of (20) are readily computed. This series converges very rapidly, and we know C2n is an integer, so we need only carry out the calculation indicated in (20) until it converges one word- size (35 bits) to the right of the decimal point. It is simple to check at the same time that C2n is indeed very close to an integer, in order to verify the computations.

4. Periodicity of the Sequences. Examination of the tables produced by the computer program shows that the unit's digits of the nonzero tangent numbers repeat endlessly in the pattern 2, 6, 2, 6, 2, 6, starting with Tz ; furthermore the two least significant digits ultimately form a repeating period of length 10: 16, 72, 36, 92, 56, 12, 76, 32, 96, 52, 16, 72, ... . The three least significant digits have a period of length 50, and for four digits the period-length is 250. These empirical observa- tions suggest that theoretical investigation of period-length might prove fruitful. Theorem 1. Let p be an odd prime, and let X be the period-length of the sequence (T„ mod p). Then p = 1 (mod 4) X = p-1, (21) 2ip - 1) p = 3 (mod 4) and

(22) Tn+x = Tn (mod p) for all n à 0 . Proof. It is clear from the recurrence relation (7) that the sequence (Tn mod p) is determined by the recurrence equation

(23) yn+i = Ayn where the vector yn and the matrix A are defined by

0 2 Tn,l 1 0 3 2 0 4 I n,1 (24) 3 • !Jn

0 0 p-1 p - 2 0 . L J n,p—1J

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For Tn¡k can contribute nothing to any subsequent value of T„ when fc 2: p. We will show below that the minimum polynomial equation satisfied by A is (25) Ap-1 - (-l)(^-1)/27 = 0 (modulo p) hence (22) is valid for the value of X given by (21). It remains to show that X is the true period-length of the sequence, not merely a multiple of the period. Accordingly, suppose Tn+y = Tn(mod p) for some positive X' ^ X and all large n. In view of (22) this congruence must hold for all n ^ 0. Let y = y y — yo', then p(Any) = 0 for all n ^ 0 where p denotes the projection onto the first component of the vector Any. But this implies n\an = 0 (mod p) for all components an of y, hence y = 0, i.e., y0 = y\' = A^'yo. It follows that yn = Ax'yn for all n ^ 0, and since the vectors y0, • ■ -, yP~2 are obviously linearly independent we must have Ax = / (modulo p). Therefore, X' is à X, and the proof is complete. It remains to verify (25), which seems to be a nontrivial identity. Clearly, the minimum polynomial of A must be of degree p — 1, since y0, • • •, yP-2 are linearly independent; therefore, it suffices to calculate the characteristic polynomial of A. Let x —On — 1) -n x —(n — 2) in- 1) (26) Dn = det

x -2 then Dn — xDn-i — On — l)nD„-2 so we have

Di = x , D2 = x2 - 1-2, D3 = x3 - (1-2 + 2-3).c, Da = x" - (1-2 + 2-3 + 3-4)a:2+ 1-2-3-4, Db = xb - (1-2 + 2-3 + 3-4 + 4-5).r3+ (1-2-3-4 + 1-2-4-5 + 2-3-4-5).r , and in general

7-s, n n—2 i n—4 n—6 , (27) Dn = X — SnlX + S„2X — SnZX + ■ • • , where

(28) Snk = 2 ai(ai + l)ß2(a2 + 1) • • -ak(ak + 1) is summed over all values 1 ^ oti

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s62= 1-2-3-4+ 1-2-4-5 + 1-2-5-6 + 1-2-6-7 + 2-3-4-5 + 2-3-5-6 + 2-3-6-7+ 2-3-7-1 + 3-4-5-6 + 3-4-6-7 + 3-4-7-1 + 4-5-6-7 + 4-5-7-1 + 5-6-7-1 (modulo 7). Let us say two terms ai(ai + 1) • • ■ ak(ak + 1) and a/(ai' +1) • • • ak'(ak + 1) are "equivalent" if, for some r and t and for all j, a, = a'(y+r)mod p + t; thus, in the above example the terms 1-2-4-5, 2-3-5-6, 3-4-6-7, 4-5-7-1, 5-6-1-2, 6-7-2-3, 7-1-3-4 are mutually equivalent. It is impossible for a term to be equivalent to itself when 0 < t < p, since this would imply ai + • • • + ak = öi+ • • • + o,k + kt, and t = 0. Therefore, each equivalence class has precisely p terms in it. When fc < Op — l)/2 the sum over an equivalence class has the form Z iai + t)iai + t+!)■■■ iak + t)(ak + t + 1) 0Si

summation may be expressed modulo pasa sum of terms of the form co&0) = cC+\) = °' since o^^"1'

SO skp 0. It follows that (29) Dp-i = x*-L + (-ly^'Xp,(j>-l)/2. - 1)! (modulo p) and an application of Wilson's theorem completes the proof of (25). Theorem 2. Let p be an odd prime, and let X be the period-length of the sequence (En mod p). Then \p — 1 , p = 1 Cmod4) X = (30) l2(p - 1) , p = 3(mod4) and (31) En+X = En (mod p) for all n ^ 1 . Proof. Make the following changes in the proof of Theorem 1 :

En,0

(32) A = Un En,i

p-1 p-1 0 J L-En,js_iJ Then the minimum polynomial equation satisfied by A is (33) Ap - (-l)(p-1)/2A = 0 (modulo p) . The proof is a straightforward modification of the proof of Theorem 1. The congruences (22) and (31) were obtained long ago by Kummer (see for example [5, p. 270]), but it was not shown that the true period-length could not be a proper divisor of the number X given by (21), (30). More general congruences given

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TANGENT, EULER AND BERNOULLI NUMBERS 671 by Kummer make it possible to establish further results about the period-length : Theorem 3. Let p be an odd prime, and let X be given by (30). Then

(34) Tn+Xvh-i = Tn (modulo pk) , n ^ fc , (35) En+Xpk-i = En Omodulo ph) , n ^ fc . Proof. Assume to ^ fc and define the sequence (um) by the rule

(36) um = i-l)^1)m/2Tn+(p-i)m , m^O. Kummer's congruence for the tangent numbers may be written (37) Akum =■ 0 (modulo pk) , m ^ 0 , k ^ 1 , where Akum denotes

Um+k — I . jUm+k-l + l o )Um+k-2 — • • • + (— l)kUm .

We will prove that (37) implies

(38) Um+pr-i = Um (modulo pr) , m ^ 0 , r ^ 1 , and this will establish (34). Eq. (35) follows in the same way if we let

t*m — V XJ *-'n+(p—l)m • Assume Eq. (37) is valid for some sequence of real numbers (not necessarily integers) uo, U\, • • • ; thus, Akum is an integer multiple of pk when fc ^ 1, but not necessarily when fc = 0. We will prove that the sequence um/p, um+p/p, um+2p/p, • • •, for fixed m also satisfies Eq. (37), and this suffices to prove (38) by induction on r. Let E be the operator Eum = um+i. Eq. (37) may be written (E — l)kum = 0 (modulo pk), and our goal as stated in the preceding paragraph is to show that (Ep — l)k(um/p) = 0 (modulo pk), i.e. (Ep — l)kum = 0 (modulo pk+1). Let f(E) = Ep-2 + 2EP-* + ■■■ + (p - 2)E + (p - 1); then E" - 1 = (E - l)(p + f(E)(E - 1)), hence

(Ep - 1)V» = Z ( * )pj(E - l)2k~jf(E)k-jum

and each term in the sum on the right is an integer multiple of p2k. Hence, we have proved in fact that (Ep — l)kum = 0 (modulo p2k), which is more than enough to complete the proof of the theorem. Note that Eqs. (34), (35) do not necessarily give the true period-length of the sequence mod pk when fc > 1; although (34) is "best possible" when p = 5 and fc = 2, 3, 4, the tangent numbers have the same period-length modulo 9 as they do modulo 3. The tangent number T2n+i is divisible by 2", so the period length of T„ mod 2r is 1 for all r. Eq. (35) is valid for X = 2 when p = 2, since Kummer's congruence (37) holds for um = En+2m. In particular, we may combine the results proved above to show that for any modulus m the sequences Tn mod m, En mod m are periodic, and the period-length divides 2

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Mathematics Department California Institute of Technology Pasadena, California 91109

1. Thomas Clausen, "Theorem," Asir. Nachrichten, v. 17, 1840, cols. 351-352. 2. S. A. Joffe, "Calculation of the first thirty-two Eulerian numbers from central differences of zero," Quart. J. Math., v. 47, 1916, pp. 103-126. 3. S. A. Joffe, "Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero," Quart. J. Math., v. 48, 1917-1920,pp. 193-271. 4. D. H. Lehmee, "An extension of the table of Bernoulli numbers," Duke Math. J., v. 2, 1936, pp. 460-464. 5. Niels Nielsen, Traité Élémentaire des Nombres de Bernoulli, Paris, 1923. 6. J. Peters & J. Stein, Zehnstellige Logarithmentafel, Berlin, 1922. 7. S. Z. Serebrennikoff, "Tables des premiers quatre vingt dix nombres de Bernoulli," Mém. Acad. St. Petersbourg 8, v. 16, 1905, no. 10, pp. 1-8. 8. K. G. C. von Staudt, "Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend," J.für Math., v. 21, 1840, pp. 372-374.

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