Problems & Puzzles: Puzzles

Puzzle 561.- Square of consecutive prime & even numbers

JM Bergot sent the following puzzle:

One notices that the squares of consecutive primes added to the squares of consecutive even numbers produces sums that are prime.

5^2 + 16^2 = 281
7^2 + 18^2 = 373
11^2 + 20^2 = 521
13^2 + 22^2 = 653
17^2 + 24^2 = 805  FAIL

Q. Can you find a run longer than four that produces primes?

Solutions came from: Torbjörn Alm, Seiji Tomita, Fred Schalekamp, Emmanuel Vantieghem, Jeff Heleen.

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Torbjörn Alm wrote:

Puzzle 561 solution: NP= 5
10837^2 + 1058^2 = 118559933
10847^2 + 1060^2 = 118781009
10853^2 + 1062^2 = 118915453
10859^2 + 1064^2 = 119049977
10861^2 + 1066^2 = 119097677

Puzzle 561 solution: NP= 5
22717^2 + 678^2 = 516521773
22721^2 + 680^2 = 516706241
22727^2 + 682^2 = 516981653
22739^2 + 684^2 = 517529977
22741^2 + 686^2 = 517623677

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Seiji Tomita wrote:

#primes         First prime  First k
4               5            16
5               29           94
6               59           54
7               1259         306
8               47           7330
9               35753        97652
10              136417       91232

Example of 10 primes.

136417^2 + 91232^2 =26932875713
136421^2 + 91234^2 =26934331997
136429^2 + 91236^2 =26936879737
136447^2 + 91238^2 =26942156453
136453^2 + 91240^2 =26944158809
136463^2 + 91242^2 =26947252933
136471^2 + 91244^2 =26949801377
136481^2 + 91246^2 =26952895877
136483^2 + 91248^2 =26953806793
136501^2 + 91250^2 =26959085501

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Fred Schalekamp wrote:

53^2 +         312^2 =      100153
59^2 +         314^2 =      102077
61^2 +         316^2 =      103577
67^2 +         318^2 =      105613
71^2 +         320^2 =      107441
73^2 +         322^2 =      109013
79^2 +         324^2 =      111217

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Emmanuel Vantieghem wrote:

Here is the best that I could find for puzzle 561:

The 10 consecutive primes  {3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007} squared and added to the 10 consecutive odd squares  {90324088435074244, 90324089637232896, 90324090839391556, 90324092041550224, 90324093243708900, 90324094445867584, 90324095648026276, 90324096850184976, 90324098052343684, 90324099254502400}  give the 10 primes
{90324088450464173, 90324089652669937, 90324090854844317, 90324092057097473, 90324093259287709, 90324094461604673, 90324095663938397, 90324096866192977, 90324098068367693, 90324099270558449}.

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Jeff Heleen wrote:

17581^2 + 125130^2 = 15966608461
17597^2 + 125132^2 = 15967671833
17599^2 + 125134^2 = 15968242757
17609^2 + 125136^2 = 15969095377
17623^2 + 125138^2 = 15970089173
17627^2 + 125140^2 = 15970730729
17657^2 + 125142^2 = 15972289813
17659^2 + 125144^2 = 15972861017
17669^2 + 125146^2 = 15973714877

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Emmanuel wrote again (Nov 2010):

I found three sets of 11 consecutive primes with the desired property :

• the first one 41549, first even number to be squared  395498534,
• the first one 648779, first even number to be squared  2079679170,
• the first prime 827639, first even number to be squared  987705646.
• I found also a set of 12 consecutive primes with the desired property :
• the first one is 1048613,the first even number to be squared is 2033153978.

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