Solve (2+1)(2^{2}+1)(2^4+1)(2^8+1)(2^16+1)(2^2+1)+1

$$( 2 + 1 ) ( 2 ^ { 2 } + 1 ) ( 2 ^ { 4 } + 1 ) ( 2 ^ { 8 } + 1 ) ( 2 ^ { 16 } + 1 ) ( 2 ^ { 2 } + 1 ) + 1$$

Evaluate

$$21474836476$$

Solution Steps

Multiply 2^{2}+1 and 2^{2}+1 to get \left(2^{2}+1\right)^{2}.

$$\left(2+1\right)\left(2^{2}+1\right)^{2}\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Add 2 and 1 to get 3.

$$3\left(2^{2}+1\right)^{2}\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Calculate 2 to the power of 2 and get 4.

$$3\left(4+1\right)^{2}\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Add 4 and 1 to get 5.

$$3\times 5^{2}\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Calculate 5 to the power of 2 and get 25.

$$3\times 25\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Multiply 3 and 25 to get 75.

$$75\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Calculate 2 to the power of 4 and get 16.

$$75\left(16+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Add 16 and 1 to get 17.

$$75\times 17\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Multiply 75 and 17 to get 1275.

$$1275\left(2^{8}+1\right)\left(2^{16}+1\right)+1$$

Calculate 2 to the power of 8 and get 256.

$$1275\left(256+1\right)\left(2^{16}+1\right)+1$$

Add 256 and 1 to get 257.

$$1275\times 257\left(2^{16}+1\right)+1$$

Multiply 1275 and 257 to get 327675.

$$327675\left(2^{16}+1\right)+1$$

Calculate 2 to the power of 16 and get 65536.

$$327675\left(65536+1\right)+1$$

Add 65536 and 1 to get 65537.

$$327675\times 65537+1$$

Multiply 327675 and 65537 to get 21474836475.

$$21474836475+1$$

Add 21474836475 and 1 to get 21474836476.

$$21474836476$$

Factor

$$2^{2}\times 479\times 11208161$$