(2 3 5 7)-fractions

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 7)-fraction must have at the most two different prime divisor. The table below of such fractions will be organized with respect to such primes or pairs of primes.

Divisor  2

2/1      and      3/2
4/3      and      5/4
8/7      and      9/8
16/15
64/63

Divisor  3

3/2      and      4/3
9/8      and      10/9
28/27
81/80

Divisor 5

5/4      and      6/5
25/24
126/125

Divisor  2*3

6/5      and      7/6
25/24
36/35
49/48
4375/4374

Divisor  7

7/6      and      8/7
49/48      and      50/49
2401/2400

Divisor 2*5

10/9
21/20
50/49
81/80

Divisor  2*7

15/14
28/27
225/224

Divisor  3*5

15/14      and      16/15
225/224

Divisor  3*7

21/20
64/63

Divisor  5*7

36/35
4375/4374



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