▁are ▁not ▁powers ▁of ▁two . ▁ ▁Eu cl id ' s ▁Element s , ▁Book ▁IX ▁ ▁The ▁geometric ▁pro gression ▁ 1 , ▁ 2 , ▁ 4 , ▁ 8 , ▁ 1 6 , ▁ 3 2 , ▁… ▁( or , ▁in ▁the ▁binary ▁numer al ▁system , ▁ 1 , ▁ 1 0 , ▁ 1 0 0 , ▁ 1 0 0 0 , ▁ 1 0 0 0 0 , ▁ 1 0 0 0 0 0 , ▁… ▁) ▁is ▁important ▁in ▁number ▁theory . ▁Book   IX , ▁Proposition   3 6 ▁of ▁Element s ▁proves ▁that ▁if ▁the ▁sum ▁of ▁the ▁first ▁ ▁terms ▁of ▁this ▁pro gression ▁is ▁a ▁prime ▁number ▁( and ▁thus ▁is ▁a ▁M ers enne ▁prime ▁as ▁mentioned ▁above ), ▁then ▁this ▁sum ▁times ▁the ▁th ▁term ▁is ▁a ▁perfect ▁number . ▁For ▁example , ▁the ▁sum ▁of ▁the ▁first ▁ 5 ▁terms ▁of ▁the ▁series ▁ 1 ▁+ ▁ 2 ▁+ ▁ 4 ▁+ ▁ 8 ▁+ ▁ 1 6 ▁= ▁ 3 1 , ▁which ▁is ▁a ▁prime ▁number . ▁The ▁sum ▁ 3 1 ▁multip lied ▁by ▁ 1 6 ▁( the ▁ 5 th ▁term ▁in ▁the ▁series ) ▁equals ▁ 4 9 6 , ▁which ▁is ▁a ▁perfect ▁number . ▁ ▁Book   IX , ▁Proposition   3 5 , ▁proves ▁that ▁in ▁a ▁geometric ▁series ▁if ▁the ▁first ▁term ▁is ▁subtract ed ▁from ▁the ▁second ▁and ▁last ▁term ▁in ▁the ▁sequence