It is unlikely that there is a simple method, not involving relatively expensive chemical analysis, for distinguishing 4N, 5N, and 6N copper specimens. Shown below is a photograph of some copper specimens I have:

The evaporating dish contains copper shot, the large plate (4.76 mm thick) is OFHC (oxygen free high conductivity) copper and the third specimen is a copper bar that is 25.4 mm by 31.5 mm cross section. The standard banana is for scale. All of these copper specimens are supposedly 'pure', but how pure is 'pure'?
The problem with determining copper purity, when the specimens are already what might be considered 'pure enough,' is that the impurities are present at low levels and their identities matter a great deal. This figure, from Edward R. Tufte, The Visual Display of Quantitative Information, Graphics Press, Cheshire, CT, USA, ©1983, p. 49, shows published copper thermal conductivities on a log-log scale. The original publication reference is given as Ref.1 (page I-244).

This beautiful figure collects results from hundreds of studies in the published literature and shows, quite dramatically, that the large majority of those studies were incorrect, sometimes wildly so. What tripped up so many researchers is that they assumed they were measuring the thermal conductivity of copper specimens that were pure enough. But they were not pure enough. And oxygen turns out to be a major culprit in reducing the thermal conductivity of copper metal. Given that we live in an oxygen atmosphere, this is a problem. As well, trace oxygen determination is not an easy thing without appropriate instrumentation and methodology.
Bottom line: you need to know what the major impurities are (or might be) and then do trace determinations of those.
References:
- C. Y. Ho, R. W. Powell, and P. E. Liley, “Thermal Conductivity of the Elements: A Comprehensive Review,” Journal of Physical and Chemical Reference Data 1974, 3 (Supp. No. 1), pp. I-244 (ISBN-13: 978-0-88318-216-1).