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There are NO patterns in primes - anywhere. If you find one, your name will go down in mathematical legend!
how i can identified and describe number patterns
There need not be any patterns at all.
Assuming that I understand you correctly, I would say that a first step would be to make a scatterplot and to examine it for patterns.
One advantage of graphing daa in general is to see patterns. Many types of graphs will show patterns. Line graphs are excellent for seeing patterns and trends.