that, for some constant c, there always exists a square-free integer between x and for positive x. Moreover, an elementary argument allows us to replace by The ABC conjecture would allow . Table of Q(x), x, and R(x) The table shows how and compare at powers of 10. , also denoted as . {| class="wikitable" style="text-align: right" ! ! ! ! |- | 10 | 7 | 6.1 | 0.9 |- | 102 | 61 | 60.8 | 0.2 |- | 103 | 608 | 607.9 | 0.1 |- | 104 | 6,083 | 6,079.3 | 3.7 |- | 105 | 60,794 | 60,792.7 | 1.3 |- | 106 | 607,926 | 607,927.1 | |- | 107 | 6,079,291 | 6,079,271.0 | 20.0 |- | 108 | 60,792,694 | 60,792,710.2 | |- | 109 | 607,927,124 | 607,927,101.9 | 22.1 |- | 1010 | 6,079,270,942 | 6,079,271,018.5 | |- | 1011 | 60,792,710,280 | 60,792,710,185.4 | 94.6 |- | 1012 | 607,927,102,274 | 607,927,101,854.0 | 420.0 |- | 1013 | 6,079,271,018,294 | 6,079,271,018,540.3 | |- | 1014 | 60,792,710,185,947 | 60,792,710,185,402.7 | 544.3 |- | 1015 | 607,927,101,854,103 | 607,927,101,854,027.0 | 76.0 |- |} changes its sign infinitely often as tends to infinity. The absolute value of is astonishingly small compared with . Encoding as binary numbers If we represent a square-free number as the infinite product then we may take those and use them as bits in a binary number with the encoding The square-free number 42 has factorization , or as an infinite product Thus the number 42 may be encoded as the binary sequence ...001011 or 11 decimal. (The binary digits are reversed from the ordering in the infinite product.) Since the prime factorization of every number is unique, so also is every binary encoding of the square-free integers. The converse is also true. Since every positive integer has a unique binary representation it is possible to reverse this encoding so that they may be decoded into a unique square-free integer. Again, for example, if we begin with the number 42, this time as simply a positive integer, we have its binary representation 101010. This decodes to Thus binary encoding of squarefree numbers describes a bijection between the nonnegative integers and the set of positive squarefree integers. (See sequences A019565, A048672 and A064273 in the OEIS.) Erdős squarefree conjecture The central binomial coefficient is never squarefree for