Ideal Waring Theorem for the Polynomial m(x3- x)/6 - m(x2- x)/2 + x

Ideal Waring Theorem for the Polynomial m(x3 - x)/6 - m(x2 - x)/2 + xAuthor(s): Alvin SugarSource: American Journal of Mathematics, Vol. 59, No. 1 (Jan., 1937), pp. 43-49Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371558 .

Accessed: 05/12/2014 20:47

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 130.194.20.173 on Fri, 5 Dec 2014 20:47:56 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/2371558?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


IDEAL WARING THEOREM FOR THE POLYNOMIAL m(x3 -x)/6- m(x2- x)/2 + x.1

By ALVIN SUGAR.

This paper is numbered in sequel to a previous paper by the author 2 in which there was obtained the second known universal Waring theorem for a polynomial with a parameter.3

THEOREM. Every positive integer is a sum of m + 3 values of P(x) = M(x3 - x)/6 + x for non-negative integers x, where m ? 16.

Since the completion of that paper, Dickson 4 has proved the ideal universal Waring theorem for n-th powers, for n > 6. This is, then, the third universal Waring theorem for a polynomial with a parameter. In this paper we shall add a fourth theorem to this list. And, furthermore, this result, as were the three preceding, is also an ideal result.

5. A universal Waring theorem obtained by a transformation. If we subject

(13) f(x) M(x3 -x)/6 -M(x2-x)/2 +x, x integral and ? 0,

to the transformation x = y + 1, we get

(14) f(x) =P(y) +1

From (14) and the above theorem, we can immediately conclude that every positive integer N > m + 3 is a sum of m + 3 values of f (x), when m ? 16. Since every integer ? m + 3 is surely a sum of m + 3 values (0 and 1 being values of f (x) ), we have proved the first part of the following theorem.

THEOREM 8. Every positive integer is a sum of m + 3 values of (13) for m > 16 and is a sum of nine values for 1 ? n? 6.5

1 Presented to the Society, August 31, 1936. Received by the Editors, October 19, 1936.

2American Journal of Mathematics, vol. 58 (1936), pp. 783-790. "The first theorem of this kind was proved by Cauchy, Oeuvres (2), vol. 6, pp.

320-353, and had been stated earlier by Fermat. It was only recently that James conjectured the existence of similar theorems for cubic polynomials, American Journal of Mathematics, vol. 56 (1934), p. 305.

4American Journal of Mathematics, vol. 58 (1936), pp. 530-535. 5It is known that nine values of P (x) suffice for 1 _nm6. See Dickson, Trans-

actions of the American Mathematical Society, vol. 36 (1934), p. 739, Theorem 12.

43



44 ALVIN SUGAR.

This theorem, however, is not an ideal Waring theorem. But, fortunately, we can obtain the ideal Waring theorem for this polynomial.

Since the constant CO of Theorem 1 was taken more than 10 greater than the Dickson-Baker-Webber constant, we have

THEOREM 9. For m ? 7, every integer ? 1012m10 is a sum of nine or ten values of (13) according as the congruence m - 6 (mod 9) does not or does hold.

6. The ideal for f (x). We list a set of the first twelve values of f (x)

0, 1, 2, a = m + 3, b =- 4m + 4, c = 1m + 5, d = 20m + 6, e = 35m + 7, f =56m + 8, g = 84m + 9, h = 120m + 10, i =165m + 11.

The ideal g (f) is the smallest valne of s for which it is true that every positive integer is a sum of s values of f. We can prove

B(f) =[(m + 1)/2] + 3 ? g(f).

For, we see that the integer

[ m + 4

(m + 3)

requires at least [(n + 1)/2] + 3 values of f(x) when m ? 5. Our next step will be to prove g(f) ? B(f) (whence g(f) = B(f)) by proving that every positive integer is a sum of B (f) values of f (x).

7. Ascension theory. If we replace x by y + 1 in (14) we get the following identity in y:

(15) f (Y + 1)-=P(Y) + If we write

d(a) =f(a + 1) -f(a),

we see by (15) that the equality

(16) d (a + 1) =-=F(a)

holds identically in a, where F(a), as we recall, was defined to be

P(a+ 1) -P(a).

From (15) it also follows that

(17) f(a + 2) = P(a + 1) + 1 > P(a + 1).



IDEAL WARING THEOREM. 45

Now from the analogue for f(x) of Theorem 2, and these two relations, (16)

and (17), we see that we can use the following theorems in making ascensions with the polynomial f (x).

THEOREM 10. Let every integer n, c < n? g, be a sum of k - 1 values

of f (x), and let a be an integer ? 0 for which F(a) < g - c. Then every

integer N, c < N ? g + P(a + 1), is a sum of k values of f (x).

THEOREM 11. Let every integer n, c < n ? c + pm + q, be a sum of

k values of f (x), and let t be a real number ? 1 which satisfies the inequality

F(3t + 1) < pmr + q. Then every integer N,

c < N? (9/2)t2(3/2)SmM

is a sum of k + s values of f (x).

8. Two Lemmas.

LEMMA 5. Every positive integer 20m + 6 is a sum of B= B (f)

values of f for m ? 17.

LEMMA 6. Every integer between 20m + 6 and 217m + 32 is a sum of

B values of f for m > 36.

In proving Lemma 5 we proceed by a method analogous to that employed in obtaining Lemma 2 and get the following set of numbers, so constructed that between each consecutive pair, B values suffice.

3a=3m + 9, b =4m + 4, a+b =5m + 7, 2a+ b -6m + 10,

6a + m- 7 7m + 11, 6a + m -6 =7m + 12, 3a + b 7m + 13,

2b ==8m + 8, a + 2b = 9m + 11, c == 10m + 5, a + c =llm + 8,

2a+c = 12mn+11, 4a+2b+m - 8 13m+12, 3a+c c 13m+14,

b+c==14m-+I9, a+b+c=- - 15m+12, 3a.+I3b+m--8 = 16m-+I13, 4b ==16m+16, 8a+2b+m-17 -17m+15, 3a+b+c 17m+18,

2b+c=18m+13, 8a+c+m-15=19m+14, a+2b+c= 19m+16, d= 20m + 6.

In order to prove Lemma 6 we construct the following set of intervals, over each of which [ (n - 1)/2] values suffice. They will overlap for m ? 36. IIence for m ? 36 we can conclude that every integer n,

(18) 20m +5 < n?23m + 12,

is a sum of [(m- 1)/2] values of f.



46 ALVIN SUGAR.

(d ==20m + 6, 21m + 3), (5b 20m + 20, 21m + 9), (a + d 21m + 9, 22m + 4), (5a + 4b 21m + 31, 22m + 12), (2a + d 22m + 12,23m + 5), (6a + 4b 22m + 34, 23m + 13).

We note that the interval (18) has been so selected that an ascension is unnecessary; for, in virtue of Theorems 10 and 11 (since Theorems 10 and 11 for f(x) are identical with Theorems 2 and 4 for P(x)), we see that the ascension made in establishing Lemma 3 from (6) is valid here. And in the future, whenever an ascension is necessary we shall be careful to select our interval and the number of values which suffice over this interval in such a way that a new ascension will not be necessary i. e. in such a way that a corresponding ascension of the previous paper will suffice.

9. The major ascensions. The following intervals have been so constructed that every integer lying in anyone of them is a sum of B - 11 values of f. These intervals will overlap for m ? 36. Hence B - 11 values will suffice from 165m + 11 to 171m + 14.

(i 165m + 11, 166m -6), (c + e + h 165m + 22, 166m + 1), (b + 3e + f = 165m + 33, 166m + 8), (a + 4d + g 165m + 36, 166m +9), (a+4c+2d+g 165m +44, 166m +13), (a+b + 2c+4e = 165m + 45, 166m + 14), (a + i = 166m + 14, 167'm - 5), (a+c+e+h 166mn+25, 167mn+2), (a+b+3e+f 166m+36, 167mI+-9), (2a + 4d+g 166rn-+-39, 167m + 10), (2a+b+d-+I4e

166rm + 44, 167rn + 13), (a + b + 3c + 2d + e + f 166rn + 49, 167m + 16), (a + 4c + d + 3e 166m +50, 16mA + 17), (2a + i =167m + 17, 168m -4), (2a + c + e + h 1 167m + 28, 168m + 3), (2a + b + 3e + f 16m- + 39, 168m + 10), (3a + 4d + g 167m + 42, 168m + 11), (3a + b + d + 4e = 16gm + 47, 168m + 14), (3a + 4c + 2d + g 167'm + 50, 168m + 15), (3a + b + 2c + 4e = 167'm + 51, 168m + 16), (2a + b + 3c + 2d + e + f 167m + 52, 168m + 17), (2a + 4c + d + 3e= 167m + 53, 168m + 18), (2g =168m + 18, 169m -1), (3a + c + e + h = 168m + 31, 169m + 4), (2b + 2c + f + g 168m + 35, 169m + 8), (a + 2b

+ 2c + d + e + g 168m + 43, 169m + 12), (a + 2b + 4c + e + g =168m + 47, 169m + 14), (4a + b + d + 4e 168m + 50, 169m + 15), (b + i 169m + 15, 10Om -4), (b + c + e + h 169m, + 26, 170m + 3), (2b + 3e + f 169m + 37, 170m + 10), (a + 2b + d + 4e =169m + 45, 170m + 14), (2b + 3c + 2d + e + f 169m + 50, 170m + 17), (b + c + 6d + e = 169m + 52, 17O0m + 18), (a + b + i




170m + 18, 171m -3), (a + b + c + e + h =170m + 29, 171m + 4), (a + 2b + 3e + f = 170m + 40, 171m + 11), (2a + b + 2c + 3d + g

17Gm + 47, 171m + 14).

Now, from the first ascension of the previous paper and from Lemmas 5 and 6, we have the following theorem.

THEOREM 12. Every positive integer is a sum of B values of (13) for 36 C m ? 1950.

We again follow the procedure of the previous paper, and beginning from an arbitrary point, construct a set of intervals such that

B-r-3 [m+1 -r

values will suffice over each, where we take

(19) r= (R -A -E-21)/2, and positive,

and e is 0 or 1 according as R -A-1 is even or odd. We begin with f (A + 1) RBm + A + 1, and our first interval is

(Rm+A+l, (R+1)m+A-2r).

The rest of the intervals are

(20) ((R+t-l)m+3B+3t-3, (R+t)m+R+t-2r),

(21) ((R + t)m+B + t-2r, (R+ t + 1)m +B + t-4r-23), (t-i . .,. 10).

In constructing the intervals (20) and (21) we take for the initial point of (20) the integer (R + t - 1)a. The value for r in (19) was obtained by requiring that r be the greatest integer satisfying

(22) T(A, t) = (R + t)m + R + t -2r f(A + 1) + ta, (t - ,. . . 10),

uniformly in t. Substituting the value for r from (19) in T(A, t) (the end point of (20)), we get

T(A, t) = f (A + 1) + ta + (10 -t) 2 +,E.

ience T(A, t) is a sum of 12 values of f(x) (of which one value will be zero if R - A - 1 is even). From this we obtain the interval (21) in the usual way.



48 ALVIN SUGAR.

An inspection gives us the information that these intervals will overlap for

m Q(A) 4R-2A-I 2(A3-4A) - 1.

Expressing r in terms of A, we have

r 1 I (A 3, _7A) - E + 21 12 ' 2

LEMMA 7. For m ? Q(A), A : 6, every integer n,

Rm + A + ?n (R + 1O)m + A + 31,

is a sum of [(ins + 1)/2] -r values of f(x).

The analogues to statements (S,), (S2), and (S3) for the polynomial now under consideration are:

(S',) For Q(A) ? m < Q(A + 1) every integer > f(A + 1) is a sum of B values, provided A ? 10.

(S'2) For mn ? Q(A), B values will suffice from f(A + 1) to f(A + 2), when A ? 10.

(S'3) For m > Q(A), every integer n, f(11) < n ? f(A + 2), will be a sum of B values.

As before (S'3) is obtained from (S'2), and (S'l) and (S'3) are sufficient to prove the following theorem.

THEOREM 13. Every integer ? f(11) is a suqm of B values of f(x) for m ? Q (10) = 639.

In proving (S'1) we proceed as we did before. From Lemma 7 and the

ascension that established (5), we have that every integer N,

(23) f(A + 1) ?N N?_C3M = 9(10)2(3/2) mr 2

is a sum of B values of f(x) for m ? Q(A). It is evident that

(24) 1O'2m10 ? c3M, for Q(A) ? m ? M (12c3)'/9.

For A > 10 (henceforth we shall require A ? 10) we have that

A3 (25) r > e1=- 16

From (4,) we see that




32 el 12 10-4All (26) -2() - -p> e2 - 2

and from (42) we see that

(27) 1Oe2 > 10-13Al8 > Q(A + 1).

Therefore M > Q(A + 1) from (23), (24), (25), (26) and (27); and we have proved (S'1). It is evident that

c3m > Mm > rnQ(A) > f(A + 2),

and therefore by (23) we have established (S'2). From Lemmas 5 and 6 and Theorems 12 and 13 we get

THEOREM 14. Every positive integer is a sum of [(rn + 1)/2] + 3 values of m (x3- x),/6 - M(x2-x)/2 + x for non-negative integers x, where m >? 36.

We embody the results of Theorems 8 and 14 in a final, recapitulative theorem.

THEOREM 15. Every positive integer is a sum of [(in + 1)/2] + 3, n + 3, or 9 values of n (x3- x) /6 n (x2 x)/2+x for non-negative integers x according as m ? 36, 35 >m m > 16, or 6 ?> m > 1.

TIHE UNIVESITY OF CALIFORNIA,

BERKELEY, CALIFORNIA.

4



Documents

Ideal Waring Theorem for the Polynomial m(x3- x)/6 - m(x2- x)/2 + x