Tuesday, September 27, 2011

Sudoku lines factorised.

Sudoku lines factorised. A. It is a demerit of Sudoku (vis-a-vis e.g. the crossword or chessproblem) that there is nothing of memorable interest in the finaldiagram, once the pleasure of solving is over. One way to mitigate thisis to factorise individual lines, each of which contains the nine digitsunrepeated. These digits can be permuted in 9! = 362880 ways, so thatthere are that many different S-numbers ranging from 123456789 =[3.sup.2].3607.3803 (A1) to 987654321 = [3.sup.2].[17.sup.2].379721(A2). We may note that all S-numbers are divisible by 9; none isdivisible by both 2 and 5; to be divisible by 11, the sums of thealternate digits must be 17 and 28; and where the middle triad is thesum of the first and third triads and its digits sum to 18, the numberis divisible by 7, 11 and 13. B. Fewest prime factors The theoretical minimum of 3 is found forall S-numbers of form [3.sup.2].p. The lowest four are: 123458679 = [3.sup.2]. 13717631 (B1) 123458967 = [3.sup.2]. 13717663 (B2) 123468957 = [3.sup.2]. 13718773 (B3) 123469587 = [3.sup.2]. 13718843 (B4) C. Most prime factors Since [2.sup.27].[3.sup.2] has ten digits,the theoretical maximum is 28. The most I have found is 18 in 918245376= [2.sup.12].[3.sup.3].[19.sup.2].23 (C1). D. Fewest different prime factors Since no S-number is a power of3, the theoretical minimum is 2 in the form [3.sup.m].[p.sup.n], as in Babove or 735982641 = [3.sup.2].[9043.sup.2] (D1) or 185742639 =[3.sup.6].254791 (D2). E. Most different prime factors Since2.[3.sup.2].7.11.13.17.19.23.29 has ten digits, the theoretical maximumis 8, but I have not found an example. I have found 64 examples of 7:55of these are divisible by 7, 11 and 13, including the lowest example127495368 = [2.sup.3].[3.sup.2].7.11.13.29.61 (E1), the roundest example283459176 = [2.sup.3].[3.sup.4].7.11.13.19.23 (E2) and the only two oddexamples 278693415 = [3.sup.2].5.7.11.13.23.269 (E3) and 746981235 =[3.sup.2].5.[7.sup.2].11.13.23.103 (E4); 8 are divisible by 7 and 11(but not by 13), including the highest example 918567342 =2.[3.sup.2].7.11.47.59.239 (E5); and one is divisible by 7 and 13 (butnot by 11), namely 537912648 = [2.sup.3].[3.sup.2].7.13.19.29.149 (E6).The 7 examples listed below came nearest to the theoretical maximum inthat the quotient left after the 6 lowest factors had been divided outwas large enough to allow for 2 more factors, but that quotient provedto be prime or, in one tantalising case, a square, as follows: 167549382 = 2.[3.sup.2].7.11.13.17.547 (E7) 382576194 = 2.[3.sup.2].7.11.13.17.1249 (E8) 475693218 = 2.[3.sup.2].7.11.13.17.1553 (E9) 541927386 = 2.[3.sup.2].7.11.13.19.1583 (E10) 657981324 = [2.sup.2].[3.sup.2].7.11.13.19.[31.sup.2] (E11) 745963218 = 2.[3.sup.2].7.11.13.19.2179 (E12) 782936154 = 2.[3.sup.2].7.11.13.19.2287 (E13) F. Highest powers The highest powers I have found for the firstfour primes are [2.sup.12] in C1 above, [3.sup.9] in 784269135 =[3.sup.9].5.13.613 (F1), [5.sup.7] in 314296875 =[3.sup.3].[5.sup.7].149 (F2) and [7.sup.5] in 129783654 =2.[3.sup.3].[7.sup.5].11.13 (F3). For the next four primes the highestpower is [17.sup.4] in 589324176 =[2.sup.4].[3.sup.2].[7.sup.2].[17.sup.4] (F4). G. Roundest numbers The seven roundest S-numbers I have foundare:(i) (i) 423579618 = 2.[3.sup.6].[7.sup.4].[11.sup.2] (G1) and 847159236= [2.sup.2].[3.sup.6].[7.sup.4].[11.sup.2] (G2) (ii) F3 above and 418693275 =[3.sup.2].[5.sup.2].7.[11.sup.2].[13.sup.3] (G3) (iii) F4 above (iv) 243918675 = [3.sup.3].[5.sup.2]:7.11.13.[19.sup.2] (G4) and249567318 = 2.[3.sup.8].7.11.13.19 (G5). It will be seen that G2 is twice G1, and there are other pairs ofS-numbers which are similarly closely related. H. Squares Table 64 in Albert Beiler's "Recreations inthe Theory of Numbers" (Dover, New York, 1966) lists 30 S-numberswhich are perfect squares, including D1, F4 and G2 above. The lowest ofthem is 139854276 = [2.sup.2].[3.sup.8].[73.sup.2] (H1), and the highestis 923187456 = [2.sup.8].[3.sup.4].[211.sup.2] (H2). The most differentprime factors is 4, shown in F4 and G2 and also in 714653289 =[3.sup.2].[7.sup.2].[19.sup.2].[67.sup.2] (H3). The largest prime whosesquare divides a S-number is 9043 in D1 above. There are no doubt other interesting aspects of the factorisationof S-numbers, and I hope that the 34 examples I have given may stimulatefurther investigation. SIR JEREMY MORSE London, England

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