Prove that #3^x-1=y^4# or #3^x+1=y^4# have not integer positive solutions. ?

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Answer 1 · 2016-09-12T20:02:50

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Case #bb(3^x+1 = y^4)#

If #3^x +1 = y^4# then:

#3^x = y^4-1 = (y-1)(y+1)(y^2+1)#

If #y# is an integer, then at least one of #y-1# and #y+1# is not divisible by #3#, so they cannot both be factors of an integer power of #3#.

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Case #bb(3^x-1 = y^4)#

If #3^x - 1 = y^4# then:

#3^x = y^4 + 1#

Consider possible values of #y^4+1# for the values of #y# modulo #3#:

#0^4 + 1 -= 1#

#1^4 + 1 -= 2#

#2^4 + 1 -= 2#

Since none of these is congruent to #0# modulo #3#, they can not be congruent to #3^x# for positive integer values of #x#.