United States 51-star flag
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The 51-star flag is a proposed design for the Flag of the United States by the U.S. Army Institute of Heraldry. It may be used if or when the United States adds a fifty-first state to the Union, such as Puerto Rico or the District of Columbia. It is also sometimes flown by proponents of New York City secession. The first known flag to have been designed was one made by Robert G. Heft, designer of the current 50-star flag.
The flag features six rows of stars, alternating nine stars and eight stars. The three rows of nine (27) and the three rows of eight (24) create 51 stars.
An alternative design of a 51-star flag (shown below) is commonly used by the Puerto Rico statehood movement, associated with the New Progressive Party of Puerto Rico, and features a circular design. The symmetry of this circular pattern is possible because 51 is the 6th pentagonal number. A similar design could be used for 52 stars (52 being the 4th decagonal number), but not 53 stars (which would be four rows of eight (32) and three rows of seven (21)).
The present 50-star design does not make use of the prime factorization of 50 (for instance, by using 5 rows of 10 stars). Rather, it uses an arrangement which can be viewed as 5 long and 4 short rows, or 6 long and 5 short columns, or, most symmetrically, as a 6-by-5 rectangle with a 5-by-4 rectangle interleaved. This would serve for any number which could be written in the form rc + (r − 1)(c − 1), where r designates the length of a long row and c the length of a long column. This includes many prime numbers such as 
, and 
.
