Pumpkin, fruit of certain varieties of squash (namely, Cucurbita pepo and C. moschata) in the gourd family (Cucurbitaceae), usually characterized by a hard orange rind with distinctive grooves. Pumpkins are commonly grown for human food and also for livestock feed. In Europe and South America, pumpkin is mainly served as a vegetable and used interchangeably with other winter squashes. In the United States and Canada, pumpkin pie is a traditional Thanksgiving and Christmas dessert. In some places, pumpkins are used as Halloween decorations known as jack-o’-lanterns, in which the interior of the pumpkin is cleaned out and a light is inserted to shine through a face carved through the wall of the fruit.

Pumpkins, which produce very long annual vines, are planted individually or in twos or threes on little hills about 2.5 to 3 metres (8 to 10 feet) apart. The fruits are generally large, 4–8 kg (9–18 pounds) or more, and yellowish to orange in colour, and they vary from oblate to globular to oblong, though some varieties are very small or feature a white rind. The rind is smooth and usually lightly furrowed or ribbed; the fruit stem is hard and woody, ridged or angled. The fruits mature in early autumn and can be stored for a few months in a dry place well above freezing temperatures. The largest pumpkins are varieties of C. maxima and may weigh 34 kg (75 pounds) or more; the most-massive pumpkins ever grown have exceeded 907 kg (2,000 pounds). Some varieties of C. argyrosperma are also known as pumpkins.

**Pumpkin nutrition facts**

Pumpkin fruit is one of the widely grown vegetables incredibly rich in vital antioxidants, and vitamins. Though this humble backyard vegetable is low in calories, nonetheless, it packed with vitamin-A, and flavonoid polyphenolic antioxidants such as lutein, xanthin, and carotenes in abundance.

Pumpkin is a fast-growing vine that creeps along the surface in a similar fashion as that of other Cucurbitaceae family vegetables and fruits such as cucumber, squash, cantaloupes, etc. It is one of the most popular field crops cultivated around the world, including in the USA at commercial scale for its fruit, and seeds.

The fruits vary widely in shape, size, and colors. Giant Pumpkins weigh 4–6 kg with the largest capable of reaching a weight of over 25 kg. Golden-nugget pumpkins are flat, smaller and feature sweet, creamy orange color flesh.

Pumpkins, in general, feature orange or yellow exterior skin color; however, some varieties can exhibit dark to pale green, brown, white, red and gray. Yellow-orange pigments largely influence their color characteristics in their skin and pulp. Its thick rind is smooth with light, vertical ribs.

In structure, the fruit features golden-yellow to orange flesh depending upon the polyphenolic pigments in it. The fruit has a hollow center, with numerous small, off-white colored seeds interspersed in a net-like structure. Pumpkin seeds are an excellent source of protein, minerals, vitamins, and omega-3 fatty acids.

**Health Benefits of Pumpkin**

• It is one of the very low-calorie vegetables. 100 g fruit provides just 26 calories and contains no saturated fats or cholesterol; however, it is rich in dietary fiber, anti-oxidants, minerals, vitamins. The vegetable is one of the food items recommended by dieticians in cholesterol controlling and weight reduction programs.

• Pumpkin is a storehouse of many anti-oxidant vitamins such as vitamin-A, vitamin-C, and vitamin-E.

• At 7,384 mg per 100 g, it is one of the vegetables in the Cucurbitaceae family featuring highest levels of vitamin-A, providing about 246% of RDA. Vitamin-A is a powerful natural antioxidant and is required by the body for maintaining the integrity of skin and mucosa. It is also an essential vitamin for good eyesight. Research studies suggest that natural foods rich in vitamin-A may help the human body protect against lung and oral cavity cancers.

• It is also an excellent source of many natural poly-phenolic flavonoid compounds such as α, ß-carotenes, cryptoxanthin, lutein, and zeaxanthin. Carotenes convert into vitamin-A inside the human body.

• Zea-xanthin is a natural anti-oxidant which has UV (ultra-violet) rays filtering actions in the macula lutea in the retina of the eyes. Thus, it may offer protection from "age-related macular disease" (ARMD) in the older adults.

• The fruit is a good source of the B-complex group of vitamins like folates, niacin, vitamin B-6 (pyridoxine), thiamin, and pantothenic acid.

• It is also a rich source of minerals like copper, calcium, potassium and phosphorus.

• Pumpkin seeds Pumpkin seeds indeed are an excellent source of dietary fiber and mono-unsaturated fatty acids, which are good for heart health. Also, the seeds are concentrated sources of protein, minerals, and health-benefiting vitamins. For instance, 100 g of pumpkin seeds provide 559 calories, 30 g of protein, 110% RDA of iron, 4987 mg of niacin (31% RDA), selenium (17% of RDA), zinc (71%), etc., but zero cholesterol. Further, the seeds are an excellent source of health promoting amino acid tryptophan. Tryptophan converted into GABA in the brain.

**Selection and storage**

Pumpkins can be readily available in the market year around. Buy completely developed, whole pumpkin fruit instead of its sections. Look for mature fruit that features a fine woody note on tapping, heavy in hand and stout stem. Avoid the one with wrinkled surface, cuts and bruises.

Once at home, ripe, mature pumpkin may be stored for many weeks to come under cool, well-ventilated place at room temperature. However, cut sections should be placed inside the refrigerator where it can keep well for a few days.

**Preparation and serving methods**

Some hybrid varieties usually subjected to insecticide powder or spray. Therefore, wash them thoroughly under running water in order to remove dirt, soil and any residual insecticides/fungicides.

Cut the stem end and slice the whole fruit into two halves. Remove inner net-like structure and set aside seeds. Then cut the flesh into desired sizes. In general, small cubes preferred in cooking preparations.

Almost all the parts of the pumpkin plant; fruit, leaves, flowers and seeds, are edible.

Here are some serving tips:

• Pumpkin can be employed in a variety of delicious recipes either baked, stew- fried; however, it is eaten best after steam-cooking in order to get maximum nutrients. In China, young tender, pumpkin leaves consumed as cooked greens or in soups.

• In the Indian subcontinent where it is popular as "kaddu or sitaphal," pumpkin is used in the preparation of "sabzi," sweet dishes (halwa), desserts, soups, curries, etc.

• The fruit employed in the preparations of pies, pancakes, custard, ravioli, etc., in Europe and US.

• Golden nugget pumpkins are used to make wonderful soufflés, stuffing, soups, etc.

• Roasted Pumpkin seeds (Pepita) can be eaten as snacks.

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My argument is that because divisors of the Mersenne number

can’t be < p if p is a prime number. Therefore if 2p +1 is a divisor of it has no divisors as p is > the square root of 2p + 1. This will therefore make 2p + 1 a prime number.Is this proof correct?

]]>**Every integer >5 can be expressed as the sum of 3 primes**

Because I find that really **beautiful**, but then I realised that this was still just a conjecture.

Whoops!

]]>The plane is infinitely extending and not on a sphere.

Antipodal points don't exist.

Lines have at least one point at infinity.

The natural units of length and area are fixed, but currently unknown; it is unknown if a natural unit of angle measurement exists.

Polar lines don't exist.

The area of a triangle is unbounded.

A representation I came up with is where lines seem to bend in parabolic shapes away from some chosen reference point, with the bending getting more extreme the farther the line is from the point; lines that go through the reference point appear straight, and how extremely a line bends away from the point is defined by the natural unit of length. There are problems with this representation though:

If you have two distinct lines, there will be two points at which they will intersect, but if you move the reference point the intersection points will also move.

A pair of perpendicular lines with at least one of the lines on the reference point intersect in one place while perpendicular lines off the reference point intersect in two places.

Line segments move around in unknown ways when the reference point moves.

The angle of intersection points might change with the position of the reference point but I don't know for sure.

“ If a Sophie Germain prime p is congruent to 3 (mod 4), then it’s matching safe prime 2p + 1 will be a divisor of the Mersenne number 2^p - 1.”

And:

“ Fermat's little theorem states that if p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p.”

However if an integer, 2p + 1, where p is a prime number, is a divisor of the Mersenne number 2^p - 1, then 2p + 1 is a safe prime and p it’s matching Sophie Germain prime.

Divisors of the Mersenne number 2^p - 1 can’t be < p if p is a prime number. Therefore if 2p +1 is a divisor of 2^p - 1 it has no divisors as p is > the square root of 2p + 1. This will therefore make 2p + 1 a safe prime and p it’s matching Sophie Germain prime.

For example 11 which is prime, (11*2) + 1 = 23. 2^11 - 1 is divisible by 23 making 11 a Sophie Germain prime and 23 it’s matching safe prime.

]]>Hi Βεν,

Nice contribution! Yes, the repeated iteration of the cosine function converges: actually, it converges to thefixed pointof the cosine function,

That makes so much sense! I also just tried

and it converged to its fixed point, 0. Tan has a fixed point but it is not attractive; sinh becomes a vertical line on 0 which means there is no attractive point, 0; cosh has no fixed point but it converges to ∞; 0 is tanh's attractive fixed point; cot has no attractive point due to it being tan with the x axis shifted; sec is related to tan because it shares half of its discontinuous points with tan suggesting it has no attractive point, and it does have no attractive point; csc is sec with the x axis shifted so it has no fixed point.]]>Welcome to the forum.

Yes, you should be doing your chem engg project. How long did this take?

forum rules wrote:

No Swearing or Offensive Topics. Young people use these forums, and should not be exposed to crudeness.

My first reaction was to delete it completely but then we lose all your posts. I think you should do the right thing and edit it yourself to something more wholesome.

Thanks,

Bob

]]>Mersenne Number= 2^11 -1

Factors = 23 and 89

2^11 -1 takes eleven goes to get to zero if I continually minus one and divide by 2. Watch what happens to 23 and 89 acting as remainders as I do the same to them.

2^11 -1

2^10 -1 First go

2^9 -1 Second go

2^8 -1 Third go

2^7 -1 Fourth go

2^6 -1 Fifth go

2^5 -1 Sixth go

2^4 -1 Seventh go

2^3 -1 Eighth go

2^2 -1 Ninth go

2-1 Tenth go

1-1=0 Eleventh go

23

(23-1)/2 = 11 First go

(11-1)/2 = 5 Second go

(5-1)/2 = 2 Third go

23+2 = 25, (25-1)/2 = 12 Fourth go

23+12 = 35, (35-1)/2 = 17 Fifth go

(17-1)/2 = 8 Sixth go

23+8 = 31, (31-1)/2 = 15 Seventh go

(15-1)/2 = 7 Eighth go

(7-1)/2 = 3 Ninth go

(3-1)/2 = 1 Tenth go

(1-1)/2 = 0 Eleventh go

89

8(89-1)/2 = 44 First go

89+44 = 133, (133-1/2) = 66 Second go

89+66 = 155, (155-1)/2 = 77 Third go

(77-1)/2 = 38 Fourth go

89+38 = 127, (127-1)/2 = 63 Fifth go

(63-1)/2 = 31 Sixth go

(31-1)/2 = 15 Seventh go

(15-1)/2 = 7 Eighth go

(7-1)/2 = 3 Ninth go

(3-1)/2 = 1 Tenth go

(1-1)/2 = 0 Eleventh go

They both take eleven goes to get to zero too.

Working backwards it can be seen that this must be the case. Starting with zero I multiply by 2 and add one continually until I get to 2^11 -1. The same must be done to the** remainders **which will then become 23 and 89 or zero, the factors of 2^11 -1.

Let the number of goes an odd number, y, takes to get to zero by continually minusing one and dividing by 2, (adding y when even) =z.

The z for 2^11 -1, 23 and 89 =11.

**Whatever z equals for y, 2^z -1 must be factorable by y. **When we use the method for 2^z -1, we never add 2^z -1 as it never equals an even number. This could potentially alter remainders for potential factors, but for Mersenne Numbers, this is not the case.

Does anyone know what “z” a composite can’t be?

]]>You can record such a parameterization of solutions.

]]>Did you want me to provide a solution or what? No square brackets (to hide) on my Kindle so you just have to close your eyes.

Also no diagram making, sorry.

This is 3D so I'll describe the diagram.

Let TB be the tower in the 'z' direction and BE be an easterly direction with the first observation at E (x direction).

Draw an oblique line NES to indicate the North/South direction (y) with ES= 42.4 ft.

As TEB = 45, BE = TB = h, the height of the tower.

And as TSB= 30, BS = root 3 h.

So in triangle BES we have BE = h, BS = root 3 h and ES = 42.4 with a right angle at E.

So it would be easy to use Pythag to get h (approx) 30 ft I think. But here's a construction for it. I have no paper nor instruments handy so the scale might not work.

Draw a verticle line BE' in the top right corner of your page. I'll make it 5cm to represent 10 ft. Construct a right angle at E' and extend the line right across the paper.

On a fresh sheet, construct an equilateral triangle sides 5 cm. Construct a perpendicular bisector so the height = 5 root 3 and set this as a radius. On the first diagram, centre B, make an arc to cut the horizontal line at S'

The triangle BE'S' is the right shape for BES but needs to be scaled up.

On E'S' produced Mark X so that E'X = 21.2 cm.

Draw XS parallel to BE' with BS' extended crossing it at S. SE parallel to S'E' will fix E and hence the correct size for triangle BES. h = BE.

Bob

]]>If I minus 1 and divide by 2 it will take p goes to get to zero.

Example:

2^11 -1

2^10 -1 First go

2^9 -1 Second go

2^8 -1 Third go

2^7 -1 Fourth go

2^6 -1 Fifth go

2^5 -1 Sixth go

2^4 -1 Seventh go

2^3 -1 Eighth go

2^2 -1 Ninth go

2-1 Tenth go

1-1=0 Eleventh go

Similarly if 2^p -1 has a factor of the form, n, it should take p goes to get to zero. Only, if minus 1 divided by 2 results in an even number, I should be allowed to add n to the number to continue. As this is what happens to n, as a remainder for 2^p -1, as each go takes place.

Examples:

23

(23-1)/2 = 11 First go

(11-1)/2 = 5 Second go

(5-1)/2 = 2 Third go

23+2 = 25, (25-1)/2 = 12 Fourth go

23+12 = 35, (35-1)/2 = 17 Fifth go

(17-1)/2 = 8 Sixth go

23+8 = 31, (31-1)/2 = 15 Seventh go

(15-1)/2 = 7 Eighth go

(7-1)/2 = 3 Ninth go

(3-1)/2 = 1 Tenth go

(1-1)/2 = 0 Eleventh go

89

(89-1)/2 = 44 First go

89+44 = 133, (133-1/2) = 66 Second go

89+66 = 155, (155-1)/2 = 77 Third go

(77-1)/2 = 38 Fourth go

89+38 = 127, (127-1)/2 = 63 Fifth go

(63-1)/2 = 31 Sixth go

(31-1)/2 = 15 Seventh go

(15-1)/2 = 7 Eighth go

(7-1)/2 = 3 Ninth go

(3-1)/2 = 1 Tenth go

(1-1)/2 = 0 Eleventh go

Both 23 and 89 take eleven goes to get to zero, as they must do, as they are factors of 2^11 -1, and 2^11 -1 takes eleven goes to get to zero also.

]]>Yes that is it. Or to make it easier it is the sum of the last two numbers in the sequence.

Hi Mcbattle, I've just finished an inductive proof of your claim: Let

, and if . Then, and

for all integers .

(the 2nd statement is needed to complete the induction step.)

Furthermore, you need

I guess the easiest way is to use the Euler-Binnet formula; with a little more work, it can be proved without it.

Regards,

zahlenspieler