4^m = 3^n +1023

To solve this equation, we can start by factoring 1023.

1023 = 3 * 341

341 = 11 * 31

So, 1023 = 3 * 11 * 31

Now, let's rewrite the equation:

4^m = 3^n + 3 * 11 * 31

We can see that the right side of the equation is divisible by 3. Therefore, the left side must also be divisible by 3.

Since 4 is not divisible by 3, we can conclude that 3^n must be divisible by 3.

This means that n must be at least 1.

Let's rewrite the equation again:

4^m = 3 * (3^(n-1) + 11 * 31)

Now, let's analyze the right side of the equation.

3^(n-1) + 11 * 31 is an integer, so it must be a power of 2.

Let's consider the possible values of n-1:

If n-1 = 0, then 3^(n-1) + 11 * 31 = 1 + 11 * 31 = 342, which is not a power of 2.

If n-1 = 1, then 3^(n-1) + 11 * 31 = 3 + 11 * 31 = 344, which is not a power of 2.

If n-1 = 2, then 3^(n-1) + 11 * 31 = 9 + 11 * 31 = 350, which is not a power of 2.

If n-1 = 3, then 3^(n-1) + 11 * 31 = 27 + 11 * 31 = 388, which is not a power of 2.

If n-1 = 4, then 3^(n-1) + 11 * 31 = 81 + 11 * 31 = 422, which is not a power of 2.

If n-1 = 5, then 3^(n-1) + 11 * 31 = 243 + 11 * 31 = 574, which is not a power of 2.

If n-1 = 6, then 3^(n-1) + 11 * 31 = 729 + 11 * 31 = 1070, which is not a power of 2.

If n-1 = 7, then 3^(n-1) + 11 * 31 = 2187 + 11 * 31 = 2518, which is not a power of 2.

If n-1 = 8, then 3^(n-1) + 11 * 31 = 6561 + 11 * 31 = 6892, which is not a power of 2.

If n-1 = 9, then 3^(n-1) + 11 * 31 = 19683 + 11 * 31 = 20094, which is not a power of 2.

If n-1 = 10, then 3^(n-1) + 11 * 31 = 59049 + 11 * 31 = 59380, which is not a power of 2.

If n-1 = 11, then 3^(n-1) + 11 * 31 = 177147 + 11 * 31 = 177478, which is not a power of 2.

If n-1 = 12, then 3^(n-1) + 11 * 31 = 531441 + 11 * 31 = 531772, which is not a power of 2.

If n-1 = 13, then 3^(n-1) + 11 * 31 = 1594323 + 11 * 31 = 1594654, which is not a power of 2.

If n-1 = 14, then 3^(n-1) + 11 * 31 = 4782969 + 11 * 31 = 4783300, which is not a power of 2.

If n-1 = 15, then 3^(n-1) + 11 * 31 = 14348907 + 11 * 31 = 14349238, which is not a power of 2.

If n-1 = 16, then 3^(n-1) + 11 * 31 = 43046721 + 11 * 31 = 43047052, which is not a power of 2.

If n-1 = 17, then 3^(n-1) + 11 * 31 = 129140163 + 11 * 31 = 129140494, which is not a power of 2.

If n-1 = 18, then 3^(n-1) + 11 * 31 = 387420489 + 11 * 31 = 387420820, which is not a power of 2.

If n-1 = 19, then 3^(n-1) + 11 * 31 = 1162261467 + 11 * 31 = 1162261798, which is not a power of 2.

If n-1 = 20, then 3^(n-1) + 11 * 31 = 3486784401 + 11 * 31 = 3486784432, which is not a power of 2.

If n-1 = 21, then 3^(n-1) + 11 * 31 = 10460353203 + 11 * 31 = 10460353234, which is not a power of 2.

If n-1 = 22, then 3^(n-1) + 11 * 31 = 31381059609 + 11 * 31 = 31381059640, which is not a power of 2.

If n-1 = 23, then 3^(n-1) + 11 * 31 = 94143178827 + 11 * 31 = 94143178858, which is not a power of 2.

If n-1 = 24, then 3^(n-1) + 11 * 31 = 282429536481 + 11 * 31 = 282429536512, which is not a power of 2.

If n-1 = 25, then 3^(n-1) + 11 * 31 = 847288609443 + 11 * 31 = 847288609474, which is not a power of 2.

If n-1 = 26, then 3^(n-1) + 11 * 31 = 2541865828329 + 11 * 31 = 2541865828360, which is not a power of 2.

If n-1 = 27, then 3^(n-1) + 11 * 31 = 7625597484987 + 11 * 31 = 7625597485018, which is not a power of 2.

If n-1 = 28, then 3^(n-1) + 11 * 31 = 22876792083321 + 11 * 31 = 22876792083352, which is not a power of 2.

If n-1 = 29, then 3^(n-1) + 11 * 31 = 68630377364869 + 11 * 31 = 68630377364900, which is not a power of 2.

If n-1 = 30, then 3^(n-1) + 11 * 31 = 205891132094649 + 11 * 31 = 205891132094680, which is not a power of 2.

If n-1 = 31, then 3^(n-1) + 11 * 31 = 617673396283947 + 11 * 31 = 617673396283978, which is not a power of 2.

If n-1 = 32, then 3^(n-1) + 11 * 31 = 1853020188851841 + 11 * 31 = 1853020188851872, which is not a power of 2.

If n-1 = 33, then 3^(n-1) + 11 * 31 = 5559060566555523 + 11 * 31 = 5559060566555554, which is not a power of 2.

If n-1 = 34, then 3^(n-1) + 11 * 31 = 16677181699666569 + 11 * 31 = 16677181699666600, which is not a power of 2.

If n-1 = 35, then 3^(n-1) + 11 * 31 = 50031545098999707 + 11 * 31 = 50031545098999738, which is not a power of 2.

If n-1 = 36, then 3^(n-1) + 11 * 31 = 150094635296999121 + 11 * 31 = 150094635296999152, which is not a power of 2.

If n-1 = 37, then 3^(n-1) + 11 * 31 = 450283905890997363 + 11 * 31 = 450283905890997394, which is not a power of 2.

If n-1 = 38, then 3^(n-1) + 11 * 31 = 1350851717672992089 + 11 * 31 = 1350851717672992120, which is not a power of 2.

If n-1 = 39, then 3^(n-1) + 11 * 31 = 4052555153018976267 + 11 * 31 = 4052555153018976298, which is not a power of 2.

If n-1 = 40, then 3^(n-1) + 11 * 31 = 12157665459056928801 + 11 * 31 = 12157665459056928832, which is not a power of 2.

If n-1 = 41, then 3^(n-1) + 11 * 31 = 36472996377170786403 + 11 * 31 = 36472996377170786434, which is not a power of 2.

If n-1 = 42, then 3^(n-1) + 11 * 31 = 109418989131512359209 + 11 * 31 = 109418989131512359240, which is not a power of 2.

If n-1 = 43, then 3^(n-1) + 11 * 31 = 328256967394537077627 + 11 * 31 = 328256967394537077658, which is not a power of 2.

If n-1 = 44, then 3^(n-1) + 11 * 31 = 984770902183611232881 + 11 * 31 = 984770902183611232912, which is not a power of 2.

If n-1 = 45, then 3^(n-1) + 11 * 31 = 2954312706550833698643 + 11 * 31 = 2954312706550833698674, which is not a power of 2.

If n-1 = 46, then 3^(n-1) + 11 * 31 = 8862938119652501095929 + 11 * 31 = 8862938119652501095960, which is not a power of 2.

If n-1 = 47, then 3^(n-1) + 11 * 31 = 26588814358957503287787 + 11 * 31 = 26588814358957503287818, which is not a power of 2.

If n-1 = 48, then 3^(n-1) + 11 * 31 = 79766443076872509863361 + 11 * 31 = 79766443076872509863392, which is not a power of 2.

If n-1 = 49, then 3^(n-1) + 11 * 31 = 239299329230617529590083 + 11 * 31 = 239299329230617529590114, which is not a power of 2.

If n-1 = 50, then 3^(n-1) + 11 * 31 = 717897987691852588770249 + 11 * 31 = 717897987691852588770280, which is not a power of 2.

If n-1 = 51, then 3^(n-1) + 11 * 31 = 2153693963075557766310747 + 11 * 31 = 2153693963075557766310778, which is not a power of 2.

If n-1 = 52, then 3^(n-1) + 11 * 31 = 6461081889226673298932241 + 11 * 31 = 6461081889226673298932272, which is not a power of 2.

If n-1 = 53, then 3^(n-1) + 11 * 31 = 19383245667680019896796723 + 11 * 31 = 19383245667680019896796754, which is not a power of 2.

If n-1 = 54, then 3^(n-1) + 11 * 31 = 58149737003040059690390169 + 11 * 31 = 58149737003040059690390200, which is not a power of 2.

If n-1 = 55, then 3^(n-1) + 11 * 31 = 174449211009120179071170507 + 11 * 31 = 174449211009120179071170538, which is not a power of 2.

If n-1 = 56, then 3^(n-1) + 11 * 31 = 523347633027360537213511521 + 11 * 31 = 523347633027360537213511552, which is not a power of 2.

If n-1 = 57, then 3^(n-1) + 11 * 31 = 1570042899082081611640534563 + 11 * 31 = 1570042899082081611640534594, which is not a power of 2.

If n-1 = 58, then 3^(n-1) + 11 * 31 = 4710128697246244834921603689 + 11 * 31 = 4710128697246244834921603720, which is not a power of 2.

If n-1 = 59, then 3^(n-1) + 11 * 31 = 14130386091738734504764811067 + 11 * 31 = 14130386091738734504764811098, which is not a power of 2.

If n-1 = 60, then 3^(n-1) + 11 * 31 = 42391158275216203514294433201 + 11 * 31 = 42391158275216203514294433232, which is not a power of 2.

If n-1 = 61, then 3^(n-1) + 11 * 31 = 127173474825648610542883299603 + 11 * 31 = 127173474825648610542883299634, which is not a power of 2.

If n-1 = 62, then 3^(n-1) + 11 * 31 = 381520424476945831628649898809 + 11 * 31 = 381520424476945831628649898840, which is not a power of 2.

If n-1 = 63, then 3^(n-1) + 11 * 31 = 1144561273430837494885949696427 + 11 * 31 = 1144561273430837494885949696458, which is not a power of 2.

If n-1 = 64, then 3^(n-1) + 11 * 31 = 3433683820292512484657849089281 + 11 * 31 = 3433683820292512484657849089312, which is not a power of 2.

If n-1 = 65, then 3^(n-1) + 11 * 31 = 10301051460877537453973547267843 + 11 * 31 = 10301051460877537453973547267874, which is not a power of 2.

If n-1 = 66, then 3^(n-1) + 11 * 31 = 30903154382632612361920641803529 + 11 * 31 = 30903154382632612361920641803560, which is not a power of 2.

If n-1 = 67, then 3^(n-1) + 11 * 31 = 92709463147897837085761925410687 + 11 * 31 = 92709463147897837085761925410718, which is not a power of 2.

If n-1 = 68, then 3^(n-1) + 11 * 31 = 278128389443693511257285776232061 + 11 * 31 = 278128389443693511257285776232092, which is not a power of 2.

If n-1 = 69, then 3^(n-1) + 11 * 31 = 834385168331080533771857328696183 + 11 * 31 = 834385168331080533771857328696214, which is not a power of 2.

If n-1 = 70, then 3^(n-1) + 11 * 31 = 2503155504993241601315571986088549 + 11 * 31 = 2503155504993241601315571986088580, which is not a power of 2.

If n-1 = 71, then 3^(n-1