boys v girls - Part 2

[MoreCoffee]

67 - yet another prime.

66

 

[MarkFL]

66

65 = 8^2 + 1^2 = 7^2 + 4^2 = sqrt(63^2 + 16^2) = sqrt(56^2 + 33^2)

 

[MoreCoffee]

65 = 8^2 + 1^2 = 7^2 + 4^2 = sqrt(63^2 + 16^2) = sqrt(56^2 + 33^2)

64

 

[MarkFL]

64

3 × 4^2 + 3 × 4^1 + 3 × 4^0

 

[MoreCoffee]

3 × 4^2 + 3 × 4^1 + 3 × 4^0

62 :rambo:

 

[MarkFL]

62 :rambo:

61 - yet another prime...6^2 + 5^2 = sqrt(60^2 + 11^2)

It is the exponent of the Mersenne prime 2,305,843,009,213,693,951 = 2^61-1.

 

[MoreCoffee]

61 - yet another prime...6^2 + 5^2 = sqrt(60^2 + 11^2)

It is the exponent of the Mersenne prime 2,305,843,009,213,693,951 = 2^61-1.

60 :smashfreakb:

 

[MarkFL]

60 :smashfreakb:

59 - another prime, an irregular prime since it divides the numerator of the 44th Bernoulli number.

 

[MoreCoffee]

59 - another prime, an irregular prime since it divides the numerator of the 44th Bernoulli number.

58 :faint:

 

[MarkFL]

58 :faint:

57 is the smallest number with exactly 10 representations as a sum of 3 distinct primes:

57 = 3+7+47 = 3+11+43 = 3+13+41 = 3+17+37 = 3+23+31 = 5+11+41 = 5+23+29 = 7+13+37 = 7+19+31 = 11+17+29

:rofl3:

 

[MoreCoffee]

57 is the smallest number with exactly 10 representations as a sum of 3 distinct primes:

57 = 3+7+47 = 3+11+43 = 3+13+41 = 3+17+37 = 3+23+31 = 5+11+41 = 5+23+29 = 7+13+37 = 7+19+31 = 11+17+29

:rofl3:

56

 

[MarkFL]

56

55 is the 10th Fibonacci number and the 10th triangular number. :woot:

 

[MoreCoffee]

55 is the 10th Fibonacci number and the 10th triangular number. :woot:

54 op2:

 

[MarkFL]

54 op2:

53 is a Sophie Germain prime, since 2 × 53 + 1 = 107 is also prime. :thumbsup:

 

[MoreCoffee]

53 is a Sophie Germain prime, since 2 × 53 + 1 = 107 is also prime. :thumbsup:

52 ---

 

[MarkFL]

52 ---

51 is the 6th Motzkin number.

 

[MoreCoffee]

51 is the 6th Motzkin number.

50 - Lawrence of Arabia

 

[MarkFL]

50 - Lawrence of Arabia

49 is the smallest number with exactly 8 representations as a sum of 3 distinct primes:

49 = 3+5+41 = 3+17+29 = 5+7+37 = 5+13+31 = 7+11+31 = 7+13+29 = 7+19+23 = 13+17+19

 

[MoreCoffee]

49 is the smallest number with exactly 8 representations as a sum of 3 distinct primes:

49 = 3+5+41 = 3+17+29 = 5+7+37 = 5+13+31 = 7+11+31 = 7+13+29 = 7+19+23 = 13+17+19

48 :humble:

 

[MarkFL]

48 :humble:

47 is prime and is the smallest number with 9 representations as a sum of 3 distinct primes:

47 = 3+7+37 = 3+13+31 = 5+11+31 = 5+13+29 = 5+19+23 = 7+11+29 = 7+17+23 = 11+13+23 = 11+17+19