I picked up this problem during a workshop but failed to figure out a complete solution. I even found some solutions:
${2^3} - 7 = 8 - 7 = 1^2$
${2^4} - 7 = 16 - 7 = 3^2$
${2^5} - 7 = 32 - 7 = 5^2$
${2^7} - 7 = 128 - 7 = 11^2$
${2^{15}} - 7 = 32768 - 7 = 181^2$
I belief, that these are all solutions, but I cannot proof it. Can anybody help?