Publication in Nature Journal last year, found at the following website:

http://www.nature.com/nature/links/011220/011220-2.html

The article stated that a 7-qubit "quantum computer" was used to factor the number 15. Shor's algorithm is described at the following website:

http://www.imsa.edu/~matth/cs299/node23.html

The description of Shors Algorithm defines the minimum number of qubits needed to factor a number.....

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Shor's algorithm for factoring a given integer 'n'

Step 2:

Pick an integer 'q' such that n^2 <= q < 2n^2

Step 4:

Create a quantum register and partition it into register #1 and register #2. Register #1 must have enough qubits to represent integers as large as q-1. Register #2 must have enough qubits to represent integers as large as n-1. The calculations for how many qubits are needed would be done on a classical computer.
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My Analysis:

Register #1 must have enough qubits to represent integers as large as q-1, which given the range defined for q is the same as saying all integers between 1 and at least (n^2) or at most (2n^2 - 2) must be represented. Binary representation of an integer is a string of 1's and 0's such that, starting from the rightmost digit, each '1' represents 2 raised to the power indicated by the column number, starting with zero:

111111 = (2^5) + (2^4) + (2^3) + (2^2) + (2^1) + (2^0)
111111 = 32 + 16 + 8 + 4 + 2 + 1
111111 = 63

If n=15, then (n^2) = 225, and (2n^2 - 2) = 448. A binary representation (as above) would require 8 bits to represent integers up to n^2, or 9 bits to represent all integers up to (2n^2 - 2). Therefore in a quantum computer register #1 would also require at least 8 qubits, possibly 9. By the same rules register #2 has to represent all integers between 1 and n-1. For n=15 this would require 4 qubits.

Therefore to factor the number 15 would require at least a 12-qubit quantum computer, perhaps a 13-qubit quantum computer. Even if register #1 operates independently from register #2, then you still need at least 8 qubits in register #1 to perform the factorization. This conflicts with the report published in Nature.

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