Problems & Puzzles: Puzzles

Puzzle 27.- Heinz Rectangles.

Let’s define a "Heinz Rectangle" with the following example:

given by Heinz in his pages http://www.geocities.com/CapeCanaveral/Launchpad/4057/primes.htm

Heinz rectangle (4x5)

5 +7+11+13+17= 53

7+ 11+13+17+19= 67

11+13+17+19+23= 83

13+17+19+23+29=101

Following the pattern, this is the bigger I have found:

(8x11)

3526741+ 3526771+ 3526781+ 3526793+ 3526867+ 3526909+ 3526931+ 3526933+ 3526937+ 3526949+ 3526987= 38795599

3526771+ 3526781+ 3526793+ 3526867+ 3526909+ 3526931+ 3526933+ 3526937+ 3526949+ 3526987+ 3526993= 38795851

3526781+ 3526793+ 3526867+ 3526909+ 3526931+ 3526933+ 3526937+ 3526949+ 3526987+ 3526993+ 3526997= 38796077

3526793+ 3526867+ 3526909+ 3526931+ 3526933+ 3526937+ 3526949+ 3526987+ 3526993+ 3526997+ 3527023= 38796319

3526867+ 3526909+ 3526931+ 3526933+ 3526937+ 3526949+ 3526987+ 3526993+ 3526997+ 3527023+ 3527033= 38796559

3526909+ 3526931+ 3526933+ 3526937+ 3526949+ 3526987+ 3526993+ 3526997+ 3527023+ 3527033+ 3527057= 38796749

3526931+ 3526933+ 3526937+ 3526949+ 3526987+ 3526993+ 3526997+ 3527023+ 3527033+ 3527057+ 3527059= 38796899

3526933+ 3526937+ 3526949+ 3526987+ 3526993+ 3526997+ 3527023+ 3527033+ 3527057+ 3527059+ 3527071= 38797039

Find a bigger one than this.

Jud Mc Cranie discover the following 9x21 Harvey Heinz rectangle at November 7, 1998. Here is his communication:

"Add 3037590000 to the numbers on the left side of the equation.

2687+2689+2741+2743+2837+2881+2917+2951+2969+2977+3007+3011+
3017+3019+3037+3043+3047+3067+3089+3101+3109 = 63789451939
2689+2741+2743+2837+2881+2917+2951+2969+2977+3007+3011+3017+
3019+3037+3043+3047+3067+3089+3101+3109+3131 = 63789452383
2741+2743+2837+2881+2917+2951+2969+2977+3007+3011+3017+3019+
3037+3043+3047+3067+3089+3101+3109+3131+3157 = 63789452851
2743+2837+2881+2917+2951+2969+2977+3007+3011+3017+3019+3037+
3043+3047+3067+3089+3101+3109+3131+3157+3173 = 63789453283
2837+2881+2917+2951+2969+2977+3007+3011+3017+3019+3037+3043+
3047+3067+3089+3101+3109+3131+3157+3173+3191 = 63789453731
2881+2917+2951+2969+2977+3007+3011+3017+3019+3037+3043+3047+
3067+3089+3101+3109+3131+3157+3173+3191+3193 = 63789454087
2917+2951+2969+2977+3007+3011+3017+3019+3037+3043+3047+3067+
3089+3101+3109+3131+3157+3173+3191+3193+3203 = 63789454409
2951+2969+2977+3007+3011+3017+3019+3037+3043+3047+3067+3089+
3101+3109+3131+3157+3173+3191+3193+3203+3217 = 63789454709
2969+2977+3007+3011+3017+3019+3037+3043+3047+3067+3089+3101+
3109+3131+3157+3173+3191+3193+3203+3217+3241 = 63789454999"

Two days later he mail me this:

"I found some more Heinz rectangles....    I also have 9x41, 9x57, and 9x97.  Do you want me to send them (they're pretty large)?"

I answer him that if somebody want those huge rectangles he could send them directly, but that if he get later other rectangles of more than 9 rows I'll be glad in showing one of them them in this pages.

***

But I have changed my mind and I have asked to publish his largest Heinz Square the following way:

The first prime, the last prime and the sum, of each and all row.

This is the 9x97 Heinz Rectangle gotten by Jud McCranie:

82114909 + ... + 82116701 = 7965228343
82114927 + ... + 82116703 = 7965230137
82114957 + ... + 82116707 = 7965231917
82114973 + ... + 82116721 = 7965233681
82114987 + ... + 82116743 = 7965235451
82115003 + ... + 82116757 = 7965237221
82115009 + ... + 82116773 = 7965238991
82115021 + ... + 82116779 = 7965240761
82115023 + ... + 82116791 = 7965242531

(each one of the 9 rows has 97 consecutive primes whose sum is a prime; the initial prime of each row is the second prime of the previous row)

Can you get a Heinz Rectangle with 10 rows? (by experience: "the row" is the harder variable...)

***

On December 2005, J. K. Andersen wrote:

A Heinz square also has the prime sums in the columns.
The first 9x9 square:

297177722629 + ... + 297177722921 = 2674599505243
297177722669 + ... + 297177722969 = 2674599505583
297177722707 + ... + 297177723079 = 2674599505993
297177722761 + ... + 297177723091 = 2674599506377
297177722861 + ... + 297177723107 = 2674599506723
297177722879 + ... + 297177723127 = 2674599506989
297177722899 + ... + 297177723131 = 2674599507241
297177722917 + ... + 297177723151 = 2674599507493
297177722921 + ... + 297177723187 = 2674599507763
------------ ------------
2674599505243 ... 2674599507763

The first 10xN rectangle with N<100 is 10x71:

45172382923 + ... + 45172384489 = 3207239244943
45172382947 + ... + 45172384501 = 3207239246521
45172382963 + ... + 45172384519 = 3207239248093
45172382971 + ... + 45172384529 = 3207239249659
45172382977 + ... + 45172384531 = 3207239251219
45172383023 + ... + 45172384541 = 3207239252783
45172383031 + ... + 45172384549 = 3207239254309
45172383041 + ... + 45172384601 = 3207239255879
45172383077 + ... + 45172384661 = 3207239257499
45172383079 + ... + 45172384667 = 3207239259089

***

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