A good prime is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

whose

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

is greater than the product of any two primes at the same number of positions before and after it in the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of primes. That is, good prime satisfies the inequality :

p_n^2 > p_ \cdot p_

for all 1 ≤ ''i'' ≤ ''n''−1, where ''p_k'' is the ''k''th prime. Example: the first primes are 2, 3, 5, 7 and 11. Since for 5 both the conditions :

5^2 >3 \cdot 7

5^2 >2 \cdot 11

are fulfilled, 5 is a good prime. There are infinitely many good primes. The first good primes are: : 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227,

251 __NOTOC__ Year 251 (Roman numerals, CCLI) was a common year starting on Wednesday of the Julian calendar. At the time, in the Roman Empire, it was known as the Year of the Consulship of Traianus and Etruscus (or, less frequently, year 1004 ''A ...

, 257, 269, 307, 311,

331 __NOTOC__ Year 331 ( CCCXXXI) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Bassus and Ablabius (or, less frequently, year 1084 ''Ab urbe condita''). The denomination ...

, 347, 419, 431, 541, 557, 563, 569, 587, 593,

599 __NOTOC__ Year 599 (Roman numerals, DXCIX) was a common year starting on Thursday of the Julian calendar. The denomination 599 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent m ...

641 __NOTOC__ Year 641 ( DCXLI) was a common year starting on Monday of the Julian calendar. The denomination 641 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe ...

, 727, 733, 739, 809,

821 __NOTOC__ Year 821 (Roman numerals, DCCCXXI) was a common year starting on Tuesday of the Julian calendar. Events By place Byzantine Empire * Byzantine general Thomas the Slav leads a revolt, and secures control over most of the Byzan ...

, 853, 929,

937 Year 937 ( CMXXXVII) was a common year starting on Sunday of the Julian calendar. Events By place Europe * A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southwards to Italy, pillaging the environs of ...

, 967 . An alternative version takes only ''i'' = 1 in the definition. With that there are more good primes: : 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101,

107 107 may refer to: *107 (number), the number *AD 107, a year in the 2nd century AD *107 BC, a year in the 2nd century BC *107 (New Jersey bus) *107 Camilla, a main-belt asteroid *Peugeot 107, a city car See also

*10/7 (disambiguation) *Bohrium, ...

, 127, 137, 149, 157,

163 Year 163 ( CLXIII) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Laelianus and Pastor (or, less frequently, year 916 ''Ab urbe condita''). The denomination 163 for this y ...

, 173, 179, 191, 197,

211 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

, 223, 227, 239,

, 257, 263, 269, 277, 281, 307, 311,

, 347, 367, 373, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 563, 569, 587, 593,

, 607, 613, 617, 631,

, 653, 659, 673,

701 __NOTOC__ Year 701 ( DCCI) was a common year starting on Saturday of the Julian calendar, the 701st year of the Common Era (CE) and Anno Domini (AD) designations, the 701st year of the 1st millennium, the 1st year of the 8th century, and the ...

, 719, 727, 733, 739, 751, 757, 769, 787, 809,

, 827, 853, 857,

877 __NOTOC__ Year 877 ( DCCCLXXVII) was a common year starting on Tuesday of the Julian calendar. Events By place Europe * Summer – King Charles II ("the Bald") sets out for Italy, accompanied by his wife Richilde and a number ...

, 881, 907, 929,

947 Year 947 (Roman numerals, CMXLVII) was a common year starting on Friday of the Julian calendar. Events By place Europe * Summer – A Principality of Hungary, Hungarian army led by Grand Prince Taksony of Hungary, Taksony campaign ...

, 967, 977,

991 Year 991 (Roman numerals, CMXCI) was a common year starting on Thursday of the Julian calendar. Events * March 1: In Rouen, Pope John XV ratifies the first Peace and Truce of God, Truce of God, between Æthelred the Unready and Richard I o ...

References

{{Prime number classes Classes of prime numbers fr:Liste de nombres premiers#Bon nombre premier