720 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxv 



E. Maillet 19 erroneously concluded that there is an infinitude of integers 

 not a sum of fewer than 128 eighth powers >0. 



A. Fleck 20 noted that in the proof by Lucas 10 it suffices to subtract at 

 most three biquadrates unless the given number is 48ra+, = 10, 11, 26, 

 27, 42, 43. For * = 10, subtract ! 4 +3 4 ; we get QN, where A^ = 4(2w-3) is 

 a ffl unless 2m 3 = 7 (mod 8), i. e., w = l+4/*. In the latter case, 



is a sum of 36 biquadrates since 4(8ju 27) is a SI. Treating similarly the 

 remaining 's, he concluded that JV 4 ^39. He found that 7V 3 ^13 by 

 employing Maillet 's 11 result and the formula, following from r 3 =r (mod 6), 



12 



E. Landau 21 proved that every definite integral rational function of x 

 of degree n with rational coefficients is a sum of 8 squares of integral 

 rational functions with rational coefficients, and gave references to related 

 problems. 



A. Fleck 22 proved that the square (cube) of every definite integral 

 rational function of x with rational coefficients is a sum of a finite deter- 

 minable maximum number, independent of the degree and coefficients of 

 the function, of fourth powers (sixth powers) of integral rational functions 

 of degree =1 with rational coefficients, i. e., linear functions and constants. 



Fleck 23 remarked that Maillet's 14 limit 192 for N& can easily be reduced 

 by about 36, but that the new limit is still far above the ideal limit 37 

 suggested by tables. To show that N 6 is finite, he used the identity 



12 



Hence 60n 3 is a sum of 184 sixth powers. Thus if m is any integer, 60m 

 is the sum of at most 184A^ 3 sixth powers. Since any integer- is of the form 

 60w+r, r = 0, 1, -, 59, we have N 6 ^184N 3 +59. 



E. Landau 24 lowered the limit for N* to 38. Setting x^=x s in (1), we 

 see that 6n 2 is a sum of 11 biquadrates if n is representable in the form 

 xl-\-xl-\-2xl, which is true if n is any odd number m. Hence 6m 2 and 

 6-16 m 2 are sums of 11 biquadrates. As above, Sk+j = 1, 2, 3, 5 or 6) 

 is a sum of three squares at least one of which is odd. Hence 6 times such 

 a number is a sum of 11 + 12+12 biquadrates. By arguments of the type 

 used by Fleck, 20 we get N 4 ^38. Except for numbers 48n+t,t = ll, 27, 43, 

 he proved that 37 biquadrates suffice. For these cases, A. Wieferich 25 

 showed that 37 suffice. Hence JV 4 ^37. 



19 Annali di Mat., (3), 12, 1905, 173, note. Error admitted in 1'interme'diaire dea math., 20, 



1913, 202. 



20 Sitzungsber. Berlin Math. Gesell., 5, 1906, 2-9. 



21 Math. Annalen, 62, 1906, 272-281. 



22 Ibid., 64, 1907, 567-572. 



23 Ibid., 561-6. To N = 192 must be added the number of units. 



24 Rendiconti Circolo Mat. Palermo, 23, 1907, 91-6. 

 26 Math. Annalen, 66, 1909, 106-8. 



