The Art of Thinking

The shifted logarithm, and the logarithmic function  Let  s t  be two arbitrary positive real numbers. The shifted logarithm from  s  to  t, symbolically LOG(s t),  is the (oriented) area of a curvilinear  trapezoid in plane xy that is bounded by the graph of function  y=1/x, and by the x-axis, and which stretches between parallel ("vertical") lines  x=s and x=t -- "oriented" means that this area is the customary geometric area when  s < t, and it is the negative of that geometric are when  s > t. Of course,  LOG(s t) = 0  when s=t. Thus, LOG(t s)   =   - LOG(s t) Now we define the logarithmic function log(t) := LOG(1 t) for arbitrary positive real number  t.  In particular: log(1) = 0 Obviously, LOG(q  s) + LOG(s  t)  =  LOG(q  t) for arbitrary positive real numbers  q s t. Logarithm as a translation from multiplication to addition Let lengths of the s...

Introduction A (2 3 5 7)-fraction is any fraction  (n+1)/n  such that  n  is a natural number (1 2 3 ...), and  n*(n+1) has no prime divisor > 7. Thus (2 3 5 7)-fractions form a subset of all near 1 fractions, see   https://whmth.blogspot.com/2024/05/near-1-fractions-and-computing-log2.html Earlier, in   https://whmth.blogspot.com/2024/05/team-2-3-5.html I listed all (2 3 5)-fractions, i.e. all fraction  (n+1)/n  such that  n  is a natural number (1 2 3 ...), and  n*(n+1) has no prime divisor > 5.  Obviously, the (2 3 5)-fractions form a subset of the set of all (2 3 5 7)-fractions. We saw that there are exactly 10 different (2 3 5)-fractions. A much more extensive table of near 1 fractions was included in a paper by D.H.Lehmer, see   https://whmth.blogspot.com/2024/05/dhlehmers-extensive-table-of-near-1.html The table of the (2 3 5 7)-fractions Either the numerator ot the denominator (or both) of a (2 3 5 ...

 To be exact, D.H.Lehmer ( in his paper received for publication on July 25, 1962 ) listed all integers  N := n*(n-1) such that integer  n>1, and no prime divisor of  N  is larger than 41. This of course is equivalent to studying near 1 fractions  n/(n-1) where n>1  is an integer. Actually, Lehmer presented also the prime decompositions of the related near 1 fractions  n/(n-1), for  n > 100000. The title of the mentioned D.H.Lehmer paper is: ON A PROBLEM OF STORMER. Furthermore, in addition to a theoretical discussion, Lehmer privided similar lists of integers  n*(n-a), where a = 2 3 4. D.H.Lehmer and Emma Lehmer were early pioneers of applying computers to Number Theory.

Introductions  The material below is original. I have created it in the year 1972. Originally, I applied my method right away to " team   2 3 5 7" while below I consider the simpler case, namely " team   2 3 5". Near 1 fractions In order to compute all three  log(2)  log(3)   log(5)   we will need three different near 1 fractions  (a+1)/a  such that  a  is a natural number, and  a*(a+1) is divisible by no prime different from  2 3 5. (Such three fractions should be in a sense independent, though they tend to be so). Thus, let's find all such near 1 fractions  (a+1)/a. Then we will select the three fractions with the largest denominators. It's clear that numerator  a+1  or  denominator  a  must be a power of  2  or. 3  or  5  while the "other floor" of the fraction can be divisible only by the other two (or one or none) of the primes  2  3  5, a...

We will pass from two fractions to three fraction. Let's start with: 5   =   (3/2)^4  *  (81/80)^(-1)  Fraction  3/2  is not impressive. A modest step toward greater efficiency (of computing log(5)) can be: 3/2  =  (5/4) * (6/5) Also, 5/4  =  (6/5) * (25/24) hence 3/2  =  (6/5)^2 * (25/24) and 5   =   (6/5)^8 * (25/24)^4 * (81/80)^(-1) Isn't this pretty good? -- maybe. The next note will present a better result, it will do so for all there integers 2 3 5 simultaneously.

 First of all, 3  =  2 * (3/2) and 2  =  (3/2) *(4/3) hence 3 =  (3/2)^2 * (4/3) but 3/2  =  (4/3) * (9/8) and finally: 3  =  (4/3)^3 * (9/8)^2 In other words: log(3)  =  3*log(4/3) + 2*log(9/8) Thus, a computation of  log(3)  is virtually as fast as that of  log(2) (see the previous note: https://whmth.blogspot.com/2024/05/near-1-fractions-and-computing-log2.html).

 Near 1 fractions are: 2/1   3/2   4/3  5/4  ... i.e. fractions  (n+1)/n  where  n  is an arbitrary natural number. When  n  gets larger, there is a tendency for a computation of  log(n) to get faster while a direct computation of  log(2)  is relatively slow. Thus, in order to speed up computing  log(2)  we may try to represent  2  as a product of near 1 fractions that have large denominators (hence numerators too). Let's start modestly: 2  =  (3/2) * (4/3) hence log(2)  =  log(3/2) + log(4/3) One should be able to compute  log(3/2)  faster than  log(2);  and a computation of  log(4/3)  should be still faster. However, it's not clear that computing both logarithms  log(3/2)  and  log(4/3) takes less time than the single direct calculation of  log(2). So, we would like to use fractions with larger denominators. There is...

 It took millennia but finally, in year 1990, mathematics got an adequate definition when on sci.math I ( Włodzimierz Holsztyński ) have stated: mathematics is the art of thinking This means that mathematics is an advanced activity such that it is a goal to itself. If a mathemarical result can be applied then it's wonderful, but nevertheless an application is not the primary goal -- the primery goal of mathematics is the high quality thinking.