On the Residues of Figurate Numbers

Annals of Mathematics

On the Residues of Figurate NumbersAuthor(s): Oliver E. GlennSource: Annals of Mathematics, Second Series, Vol. 25, No. 1 (Sep., 1923), pp. 57-70Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967728 .

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ON THE RESIDUES OF FIGURATE NUMBERS.*

BY OLIVER E. GLENN.

1. The elementary notion of figurate numbers as integers of a sequence whose nth term is the sum of n terms of another sequence, all of the numbers of the first series employed being unity, is extended in this paper according to the following definition. Let S be any series of integers which proceeds according to any fixed law

S(ae): al, a21 ...ra2 ... XY

and let these integers be considered as the initial numbers, respectively, of appropriate orders of differences in a table of differences. This table will be triangular in form and infinite in extent, viz.,

* * . * * * * * * * * * * * * * * * * x . .

Ca4 as+ a4 Cc *+ 2 8cis+-4 a,+3a2+33as+ "4

Pi: as 0Cc2 + CIs a,-2 a- + as

a2 C1 + a2

a1

The general element, the one in the uth column and the vth row, is

(11) (-) (uoi) awl4+1 + (Uyi) ar-uW2+ + a

* Presented to the American Mathematical Society, April, 1922; December, 1922. Cf. Dickson, History of the theory of numbers, vol. 1, chap. 9, vol. 2, chap. 1; also papers listed in index of vol. 1 under "Differences." The relation between these and the present paper is remote except, in connection with ? 6, cf. Dickson, Annals of Math. (1), vol. 2 (1896), p. 7&, and Quart. Journ. Math., vol. 33 (1902), p. 381, Glaisher, Quart. Journ. Math., vol. 30 (1899), p. 361, Carmichael, On sequences of integers defined by recurrence relations, Quart. Journ. Math., vol. 48 (1920), p. 343.

57



58 0. E. GLENN.

which will sometimes be abbreviated

U (at'-ully av'-U+2 *a' V l~~~~~t~ V

We are to study the properties of the figurate numbers P1 by means of the plexus P of residues of the numbers in P1 with regard to the modulus P, a prime. Reference is continually made to interesting particular cases of P and to geometric aids in the study of the numbers of the plexus.

When the initial series S(a) is

(1) S: 5n1+4, 5n2+3, 5n8+2, 5n4+1, 5n5, 5nc+4, 5nr4+3,.

where n1, n2, ... are arbitrary positive integers and p) 5, P takes the following form:

3 2 1 1 0 1 4 2 2 0 '2 3

4 4 0 4 1 3 3 0 3 2 1

0 1 4 2 2 0 2 3 4 4

1 3 3 0 3 2 1 1 0

2 0 2 3 4 4 0 4

3 2 1 1 0 1 4

4 4 0 4 1 3

0 1 4 2 2

1 3 3 0

2 0 2

3 2

4 Fig. 1.

The zero elements form the vertices of a quadrangular net. There is a general theorem of which this is an example. Consider also another example noting especially the configurations of the elements of the plexus which are congruent to zero modulo p(= 3):



ON THE RESIDUES OF FIGURATE NUMBERS. 59

S: 3nj, 3n2+1, 3n3+2, 3n4+1, 3n1+2, 3 n6, 3n7+2,

3n8, 3n9+1, 3n10, 3nj1+1, 3nj2+2,

0 2 1 i2 2 1 2 0 0 0 1 '2 1 1 2 1 0 0 0 2 1 2 2 1 2 0 2 2 1 0 2 1 1 0 0 1 1 2 0 1 2 2 0 0 2 2 1 0 2 1 1 0 2 2 2 2 0 2 0 1 0 1 1 1 1 0 1 0 2 0 2 2 2 2 0 2 0 0 0 1 2 1 1 2 1 0 0 0 2 1 2 2 1 2 0 0 0 1 1 0 0 1 1 2 0 1 2 2 0 0 2 2 1 0 2 1 1 0 0 1 2 0 1 0 1 1 1 1 0 1 0 2 0 2 2 2 2 0 2 0 1 1 1 2 1 0 0 0 2 1 2 2 1 2 0 0 0 1 2 1 1 0 1 2 2 0 0 2 2 1 0 2 1 1 0 0 1 1 2 0 1 1 0 1 0 2 0 2 2 2 2 0 2 0 1 0 1 1 0 2 1 2 2 1 2 0 0 0 1 2 1 1 2 1 0 2 2 1 0 2 1 1 00 1 1 2 0 1 2 2 0 2 2 2 2 0 2 0 1 0 1 1 1 1 0 2 0 0 0 1 2 1 1 2 1 0 0 0 2 1 0 0 1 1 2 0 1 2 2 0 0 2 2 0 1 0 1 1 1 1 0 1 0 2 1 1 2 1 00 0 2 1 2 2 0 1 2 2002212 2 1 0 1 1 0 1 0 2 0 2 2 0 2 1 2 2 1 2 0 2 2 1 0 2 1 1 0 2 2 2 2 0 2 0 0 0 1 1 0 0 1 2 0 1

0 Fig. 2.

2. Equivalences. If we delete from P the series S(a) and the first row of residues parallel to the hypotenuse, what is left is a plexus of figursate numbers which proceeds by the evident additive process from the second column of P as- an initial series S(at'). Similarly b~y deleting the first r additive series S(a), S(a'). ... S(cctt-1)) and the first X rows parallel to the hypotenuse



60 0. E. GLENN.

we obtain a plexus P which can be regarded as having originated from S(a(r)). WVe say that P' is additively equivalent to P.

The plexus P() derived from the sequence S(zca), z being a fixed integer, is defined to be in multiplicative equivalence with P. The null elements (mod. p) of P and P(t) are in identical positions.

3. The sieve of Eratosthenes.* The method in the theory of numbers known by the name cribrum Eratosthenis, fundamental in researches on the distribution of primes in the linear sequence of positive integers, can be extended to two dimensions in connection with the problem of the distribution of primes in a table P, of figurate numbers.

Form the plexus P for each prime 2, 3, 5, . . ., superimposing these residue tables upon each other in such a way that all residues modulis 2, 3, 5, ... of any element e of P, are written within one square or mesh of P. To find all prime elements e of P1 less than a number 1, proceed up the series of primes 2, 3, 5, ... until the first remaining number directly following the prime whose plexus was last formed has its square > 1. Then every number of P1 less than I whose mesh contains no zero is a prime. Every number > I whose mesh contains no zero is at least relatively prime to all prime numbers used in the formation of the sieve.

Consider the series formed by assigning to each number ni in (1) the value i of its subscript:

S(n): 3, 7, 11, 13, 17, 18, 23, 24, 28, 30, 34, 38, 40, 44, 45, 50,.

The complete list of primes which occur among the 136 numbers which make up the first 16 rows of the plexus P, whose initial series is S(n) is the following:

(+)- = (4:)3 7 - () =-( =13, (-) -- 17, () 23,

(7-) - (1)18+(1)23 - 41, (+) = (0)23?( )24 = 47,

-_ ()44+(1)45= 89, (4) = ()4+()44+()4 173

(15) - (4)34+ )38 + ()40 +()44+ 4)45 = 647,

* Eratosthenes, Third century B. C. Tschebyscheff (Theorie der Congruenzen, in Russian 1849, in German 1889, p. 4) gives the dates 276 (or 275) to 194 B. C.




(4) (O) 18 + 41 23 + 1 2)4 + 73 2?8

+ 74 30 + 7a 34 + 16 38 + (7) 40 3733,

(+) = o31)+2l+3l+zl + ( 18 + (6) 23 + 24 + 8 = 28-4157.

The numbers along the diagonal of such a table increase in magnitude

rapidly; for example, (14) = 191381 (= 97 .1973). In the formation of the

sieve, however, only residues are written. 4. Drift lines. Suppose az to be the element of general position of the

series S(W) and draw a straight line through at and the element number i-I of the third column. The elements which this line intersects when it is pro- duced to the opposite bound of the plexus are the following:

(2) (2i+r) = (2 ai-r + (21 )ai-r + + (2)a (T-0. ... S i -1).

If these numbers are all congruent to zero modulo p, the line is said to be a drift line.

Assume that the series S(a), reduced modulo p, is periodic, a complete period extending from ai to a+28 inclusive. Assign to the series, moreover, the property (3) ai+(2s+1)r+j + ai+(28?+1)rj = O(mod. p),

= 0 . ., s; r = 0, ..., A. Then aj+(2, 0)7-(mod.p), (p >2), and P1 is defined to be skew-symmetrical modulo p.

THEOREM. If the plexus P1 corresponding to a chosen series S(a) is skew- symmetrical modulo p, it is divided into strips of finite length and equal width by parallel oblique drift lines.

For, since (% = (2-) (a = 0, 2T), we have

( 2+41) 0(mod.p), (r = 0, ..., i+(2s+1)r-1),

on the line determined by each number ai+(2s+1)r.



62 0. E. GLENN.

THEOREM. Midway between two consecutive drift lines of P1 (modulop), intersecting zero elements of 8(a), there lies a drift line (secondary) which passes between two elements of S(a) but intersects a zero element of the first additive sequence S(a').

The elements of S(a) included between drift lines number r + 1 and r + 2 are

ai+(2stl1)r-+j ai+(28+1)r+2^ * * *-I ai+f2s+1) (r+1)-1b

whereas the elements of S(a') lying between the same pair of drift lines are represented by the following table:

a+(2e+1)r I a~+(2s+1)r+1 ...

| 1 as(2s+l) (r+1l1 l

|ai+(2+l)r+l | ai+(28+l)r+l + ai+(2f9+1)r+2 . ai+(2s+l)(r+l)-l + ai+(2a+1)(r+1)

Hence we have

az+(2S+1)r+s-j + ai+(28+1)r+s+j -- ai+(2s+l)r+8-j + ai+(28+l)r74 -j+l

+ ai+(2s+l) r+s+j + Gi+l2+)r+8+j+l 0 (mod. p).

The latter congruence is an immediate consequence of (3). Thus the sequence S(a') possesses the property of skew symmetry, the zero elements (analogous to ai+(2,,+l>,) being a+(2l),.?+ = 0 (mod.p). The plexus P", con- structed from the first additive sequence S(a') and additively equivalent to P, has drift lines which intersect the null elements a?+(28+l)r+&. Moreover, since the arithmetic mean of i + (2 s + 1) r and i + (2 s + 1) (r + 1) - 1 is i+(2s+1)r+s, the drift lines of P" bisect the intervals between those of P. This is what we were to prove.

5. Special sequences. Let e be any least residue modulo p and consider the following series SI (a)

81 (a): pni + e, pn2+ 2e, pns + 3e, *,pnt+rQ, **X

where ni, n2, ... are arbitrarily selected positive integers. The plexus B, corresponding, is the most general multiplicative equivalent modulo P of the plexus derived from the sequence of positive integers, i. e.

S(a): 1, 2, ...,.l. , 1'- 1,O P+I, ... . 2p-1,O, 2p+1, ...c




Hence R is skew symmetric. We have for additive sequences

S(a'): 3, 5. ...,p, ..., 2p-1, 2p+, 3, 5, ...

3p? * p * -4 1, 4p + 1, 3, . . .c?

S(a"): 4.2,4.3, .. ., (p -1),O,4(p+ 1), ... 4(2p-1),0,4(2p+1).... c.

THEOREM. The null elements (mod. p) of the plexus R are the vertices of a qu-adrangular net each mesh being a parallelogram.

This theorem follows from the fact that the only zeros (mod. p) are those upon drift lines and these are at the vertices of a quadrangular net, two columns in width and i (p + 1) rows in length, whose sides are drift. lines and oblique transversals; facts which are evident from the sequences S(a'), S(a").

6. Factors of combinatory numbers. Not only is the divisibility of figurate numbers by a prime p conditioned by that of the combinatory ,Cr, but the numbers (2 T) (r = 0, ..., i-1), themselves, are found as elements upon the line of skew symmetry in P in the case where as = I (mod. )) and all other numbers of S(a) are congruent to zero (cf. (2)). Most of the essential theory of the present paragraph is contained in the following two theorems. THEOREM. If any number of the set (I +?f) (,= ... p-1; j -= 1 **...2

pa?+,A), is zero modulo p, being divisible by Pr and not by 1i7', then the element +) is divisible by pr+@ and not by vr+J+1. Moreover (rPa?)

(r 1, p. . -, -1) is divisible bypr and not by pT+l. The number N (rp? +') may be written

(,rpa+Ad) (rpa +A1)... (r r p 1)...(rpaP)... p,-2p)...(rpa np) ... 1V2...(\p-1)p(p+l)...2p-1) 2p(2p+1)...(np-1)np(np+1)...(np+,f+l)... Be

In B the multiples of .p occur in periods of p. The first 8 + 1 factors in A contain the factor pa. Let the rest be regarded as forming cycles of period ) and make the kth cycle in the numerator correspond to the kth in B. Then the highest power of p in the factor kp in B is canceled by that in rpa -lp in the corresponding cycle in A and y = a. Any one of a number of excep- lions can occur if the last cycle in B is incomplete with respect to the above correspondence so that some factors p must be canceled by factors of pa from the first A + 1 factors in A. Then y = a - s < a. It is not necessary to trace these exceptions in greater detail for, whatever exceptional case



64 O. E. GLENN.

presents itself, it is certain that it will be repeated, and result in the same value of s in the last cycle of the denominator of

a4-+ (rp 8-~) ... (rPa+J-j)r-Pa+8T... (,,a+J$p (ra m-21)) ..(rpa+ n).

1 * 2... (p-1),v(p+ 1)... (2.p-1) 2p(2p+ )... (np-1)np(nv+ 1)... (np+,#+1)...

Thus, in all cases, N' is divisible by pr7+ (r = a - s) and not by pr+J+1. THEOREM. If the highest power of p which divides N is par, then the highest

power of p which divides N - (_Jr4+p) is Pr (r = 1, p * 1; s = O.*. *, r; j-? rpa+ Spa; a-1 2 S a)

We have N = p?' B,

._z(,pa+,d-jX,:pa+#,X-j) ... (rpa-a.p)(rlva-ap-).(b al).(p+@js 1

. R(j +f1) (j+ 2) ... (j+ ap+2+#i-j) (6+ ap + 3) (It+ ap+4) ... (j + sp)

Inasmuch as the number of factors in the numerator and denominator of the multiplier of pi B in N1u is Spa, the first factor in this numerator which is divisible by p, viz., rpa - ap, is of such a form that j ? (t-,) = ap a= p, where a1 is prime to p, and, since t < p, we have t -, < p. Hence the power pG in the denominator factor j+ (t -,) cancels the highest power of p, viz., pA, in the corresponding factor rpa - ap in the numerator, and the same comment applies in the case of the other factors rpa - (a + 1) p, rpa - (a + 2)p . .., which are divisible by a power of p. Thus the multiplier of prR in N1 is incongruent to zero modlo p, which proves the theorem.

The plexus Q of positive integers, each less than the prime p, obtained by reducing the numbers of Pascal's triangle according to the modulus p possesses properties of symmetry that are partly exhibited by the arrangement of the numbers of Q in geometric forms. Supposing Q to be written as an equilateral triangle of indefinite length with 1, 1 as the first row, 1, 2, 1 as the second row and so on, there is lateral symmetry with reference to the median and various periodic recurrences, vertical as well as lateral.

In row number p the end elements are 1 and the other p -1 are zeros forming the base of an inverted equilateral triangle of zeros with vertex on the median. These triangles recur in horizontal rows, two in the second row with bases in row number 2p, three in the third row with bases in row number 3p, and so on to infinity. In general, row number pr is a row of pr 1 zeros (end elements 1) the base of an inverted equilateral triangle of pr(p"-1)/2 zero elements forming the first of a series of rows of triangles of these dimensions, two triangles in the second row with bases in row




nullmber 2 pr of the plexus, three in the third row with bases in row number 3p/, cand so on to infinity.

It follows from the two theorems of the present paragraph that, within an arbitrarily chosen triangle of zeros in the cycle of rows from row number pr

of Q, to row number pr+l, there are triangles of dimensions p 1- that are submerged to the depth r, that is, their elements are also elements of zero triangles brought forward from all previous cycles. We shall say that the depth of submergence w of a chosen zero element of Q is the order of that zero (010). There results the following:

THEOREM. If w is the order of a chosen zero element in the plexus of residues of Pascal's triaknqle reduced modulo p, a prime, then the corresponding combinatory number is divisible by p10 and not by pl+l.

By means of formulas involving parameters of restricted numerical variation* we can delimit the triangle of general dimensions and general position whose elements are all zeros of order to and thus obtain a determination of all combinatory numbers which are divisible by po0 and not by pwt'. For this purpose we write the following formulas each one of which represents a set of combinatory numbers belonging to the totality found in the binomial expansions from (a + b)p to (a + b)Pr+, or, as we shall say, belonging to the cycle (1r, pr+l):

((Prl + tl)p 4-j \ i+j /

(I) (r 1, . .. x; t1 = 0, ... ,pr-l (p-i)-l; j = 0, .... 2;

i (S-1)p +l, .. . , sp-1-j; S= 1, ,, r- +tl),

+ t2) +i)

4it'

(s -1 ; 12 G,, pr-2( p 1)-1;i ? = j a) 2

(II)

i (s- 1)p2+1,,, sp2-1 -j; s= 1 , pr2+ t2),

{(pr--8 + t8)p3 +j i+j

(I -, . * ; ts O. ,, pr-3 (P 1)-1; j-0 . O.. p-12w;

(Ill)

i =(S-l)p3+ 1, ...... sp93-l-j s =-1, ... ,.. p'--+ t)

* Tmns. Amer. Math. Soc., vol. 20 (1919), p. 155.



66 0. E. GLENN.

/1+ trr + j

(XR) i+j /

( = 0.... X ; t 0,... ,p-2; j= 0,... - 2; i (s-l)p'+ 1. ,, , Sp 1 -j; s 1. ..1 + t).

The formula (I) represents numbers which are in triangles brought forward from the first cycle, formula (II) numbers in triangles from the second cycle, and so on. The previous results establish the following theorem.

THEOREM. Combinatory numbers of the r-th cycle (pr, pr+1) which are divisible by pW and not by pW+l are precisely and only those which are found in iv and only in tw of the sets (I) ..., (R).

Thus the desired numbers are obtained by forming the greatest common subset of each group of w sets selected from (I), (II), . . ., (R), and combining these subsets. For example, we find as the complete set of numbers of the second cycle (32, 38), which are divisible by 32 and not by 38, the following:

(12) (12) (13) (1) (15) (16) (18) (8) (19) (18) (18) (19)

(18) (18)s (lo(19 8l (19 (18) (18) (19\ (18) (18 (19 7 \81\ 8 } 101 \11 \11 \131 14 \141 16 \171 \171

21\ /21\ {228 (21 21\

228 (21\ (22\ (21\ (21 (21 (22\ 41 5 5 (2) 7 ()8 1 8 \13/ \141 \141 \16/ \17/ \171

(24) (24) (25) (24) (25) (24) 7 8 ~8 1 16 2471 \17

7. The period parallelogram. The additive sequence number 2x of the plexus R of paragraph 5 is congruent modulo p to e S(2x) where

S (2x): 4x (x + 1), 4x (x + 2), 4x (p -1), 0, 4x (p + 1),.

Hence, withv p 2, since by Fermat's theorem 2P1 = 1 (mod.p1), the (p - 1)-th additive sequence is congruent to the initial series S, (a), so spaced that it




begins with the term which is congruent to [4 (p -1) + 1] e and has its first zero on the first drift line. Therefore, with reference to the modulus p, not only the original sequence St (a) but also all additive sequences in their turn repeat throughout the plexus R in cycles of p. It is also evident that the configuration of residues contained between any two consecutive drift lines is identical with that between any other consecutive pair. Hence the entire plexus is a net formed by parallelograms of identical content the bounds of each being a pair of consecutive drift lines and two consecutive recurrences of the initial series S, (a). This parallelogram, here and in the general theory to follow, is designated as the period parallelogram of the plexus of figurate numbers. If we delimit by formulas the period parallelogram of general position within the plexus, we have a complete analytical description, or, as we may say, a definitive residual evaluation of all numbers of the infinite plexus.

The initial number at on the nth drift line is Ap- = np ( O(mod. p), and that of the (n 4- 1)-th drift line is a(,+I)p = (n + l)ep - 0 (mod. p). The initial series S1 (a) repeats as column number 1 + m (p - 1) and next again as column number 1 + (m + 1) (p.-. 1). Hence the four corner elements of the period parallelogram of general position in R are the following:

D('rn (P- +1 B : ( (r +1) (p-i)+ 1\ ki M(pV- ) +(n + l)] /'J(m+1l)(p-1)+(,n+1)l9)

l. l n(p 1) +1 M + l) ( l(P- 1) + I (p-1)+np I' \i(m+1) (p--l)+npP

These formulas may be written explicitly be means of (1j). Thus

( i m (p 1) + 1 p = ) ( O (n p Sm 'p )e

+ m(P1 )(np-rn0(p-1)+ l+

+ im(P-1) (np?+ m( p-1))e (mod.p).

A particular case is added showing the entire period parallelogram for p -5, e= 4 (cf. fig. 1).



68 0. E. GLENN.

004 O 4 1

DO 1 4 9 2 1 3 3 0 3 2 0 2 3 4 3 2 1 1 OB 4 4 0 0

Fig. 3.

The definitive residual evaluation of this case of the plexus R is given, therefore, by the following 22 formulas:

4 M+1 , ( 4mn+1 41 ( 4m -1 = 3(mod. 5),

( 4r 51 in1 2,n4m+1 2 4m +1 5 _ ) 2 ( (0 (mod.5), 2nz+5n+3 2m+5n+4 2ms+5n+5'

4m +2 4 4m+2 )12, 4m+2 )-(mod.5), 2m+5n+1 2rn+5n+2 2 2rn+25n+3'

(4" _ 2 4m m+2 2 1 4m+3 \

2m+5n+41 ' 2n+5n+5 , \m+5n+1 O(mod. 5),

2 mn + 42 in ( -i +--; 2, 2+ ? )?-+ 3(mod.5), (4rn +3 I4r)n +3 __ I4m+3

( 2n+5+ )_4 (r 5nl+6 )_1 ( +_ 3l(mod.5), 2,nt+5n+ 4' 2rr+5n+6 2m+5n+2 (md5)

( 4m+4+) 3 ( - _O (4m+$ ) = 2(mod.5), 2m+5n+3 .17+5nl 4 2 m+a5n+ 5

(4m+4 _

(2m +5n +6!) - 4 (mod. 5),

where naO. 1.... c n - 1, 2, ...; and ai = ie(mod.p) as in Si (a), paragraph 5.

8. General theory. The elements of S (a) which are included between two consecutive drift lines of a skew-symmetrical plexus P form a permutation,




involving repetitions, of the p least residues modulo p, and the same is true of S(a'), S(a"), ... Moreover, each element is a polynomial in the quan- tities a,, a2, ... of the type

(41) cc C1 ar?- + C2 fr+2 + * + C8s Ctr+S, (r a fixed integer),

where the coefficients cj are residues of p). Since the number of different sets (cl, c2, . .., coo) is finite, we have the following theorem.

THEOREM. Every skew-symmetricalple-xus P1 is congiruent modulo p), a prime, to a net of period parallelograms of finite dimensicns and idenitical content.

To make a more explicit determination of the width of this parallelogram it is necessary to deal with particular series S(a), as was done above in the case of Sl (a). If we assume. for S(a), any sequence whose residues are the cyclic permutations of the p residues repeated in the order

(4) (O.~ 1 2, ... p -1), (1, 2, ~...p -1, 0), (2, 3, *. - Iv 0!? 1),

then the width of the period parallelogram is p (p - 1). The null elements upon drift lines do not constitute the totality of zeros

in P. The rest are arranged, in general, in polygons forming a set of geometric figures which recur with the recurrence of the period parallelogram. The reason for this symmetry is involved with the combination of the symmetry inherent in the drift line theory with that of the null element configurations in Pascal's triangle reduced modulo p (cf. ? 6). It is difficult to obtain a general analytical description of these null element polygons but in a particular case (cf. fig. 2) they present a beautifully variegated symmetry.

The period parallelogram of the plexus of figure 2, which is an example of the general case (4), is subjoined.

00C

0 0 2 D 0 1 0 2 0 0 2 1 2 ') 3 I'2 2 2" 1 0 l2 1 1 0 2 2 2 2 0 2 2 0 0 0 2 1 1 0 0 1 2 0 2 0 1 0 1 1 1 1 2 1 0 0 0 B 0 1 2 2 0 1 1 0

Fig. 4.



70 O. E. GLENN.

The six null elements, (mod. 3), forming the enclosed triangle of zeros are

( 6n- 4 ) ( 6n4 _ \9m+3n- ' \ 9m+3fn 8,

./ 6n- 3 6 6n - 2 9m + 3 -7), (9m $ + 3n ) ( 6n-3 7) ( 6n-2 9mn+3n-"- , 9m.+3n 7I,

(m, = 1,...

THE UNIVERSITY OF PENNSYLVANIA.



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On the Residues of Figurate Numbers